Belajar tentang logika matematika, yuk! Mulai dari pengertian kalimat terbuka, pernyataan dan negasi, serta pernyataan majemuk (konjungsi, disjungsi, implikasi, dan biimplikasi).
—
Teman-teman, apa yang kamu bayangkan ketika mendengar tentang logika matematika? Kalau kamu murid laki-laki, mungkin akan bingung dan bertanya, “Kok matematika pakai logika segala?”. Sementara sebagian perempuan akan berpikir, “Logika itu mah urusan laki-laki! Perempuan tuh pakenya perasaan…” Hmmm…
Di dalam ilmu matematika, kamu juga dapat mempelajari logika. Buat apa? Tentu aja, buat mengasah otak kita dalam penarikan kesimpulan-kesimpulan. Jadi, ke depannya kita tidak asal menduga sesuatu. Tidak ada lagi deh, kalimat “Kamu bilangnya mau jemput jam 10. Kok telat? Pasti JALAN SAMA MANTAN YA?!”
Nah, dalam materi logika matematika, kita akan sering menemukan istilah-istilah, seperti negasi, konjungsi, disjungsi, dan lain sebagainya. Di artikel Matematika kelas 11 kali ini, kita akan bahas secara mudah dan ringkas, ya. Yuk, perhatikan secara seksama!
Pernyataan dan Kalimat Terbuka
Coba kamu perhatikan gambar berikut!
Hayo, dari gambar di atas, tahu nggak bedanya pernyataan dan kalimat terbuka? Pernyataan adalah kalimat yang bisa benar atau bisa salah. Sementara kalimat terbuka adalah jenis kalimat yang belum diketahui kebenarannya. Sehingga, untuk menentukan benar atau salahnya, kita perlu pengamatan lebih lanjut.
Baca Juga: Cara Mencari Determinan dan Invers Matriks
—
Sudah tahu belum, di Aplikasi belajar Ruangguru, ada fitur Drill Soal yang berisi kumpulan contoh soal latihan beserta pembahasannya, loh. Pas banget kan buat mempersiapkan diri kamu dalam menghadapi ujian nanti. Yuk, klik banner di bawah ini untuk coba fitur Drill Soal!
Kalau kamu masih bingung seperti apa itu contoh pernyataan, berikut adalah salah satu contohnya:
- Indonesia Raya adalah lagu kebangsaan Indonesia. (pernyataan benar)
- Bika ambon berasal dari Ambon. (pernyataan salah)
Di sisi lain, contoh dari kalimat terbuka adalah sebagai berikut:
- 12x + 6 = 91 (pernyataan ini dinamakan kalimat terbuka karena masih harus dibuktikan kebenarannya. Apakah benar 12x jika dijumlahkan dengan 6 akan menghasilkan 91?)
- Maaf ya, aku semalem ketiduran. Hehehe. (Pernyataan ini dinamakan kalimat terbuka karena masih harus dibuktikan kebenarannya. Apakah benar dia semalem nggak bales chat karena ketiduran? Atau emang males aja chat sama kamu?)
Nah, setelah mengetahui apa itu pernyataan dan kalimat terbuka, sekarang kita lanjut pembahasan mengenai ingkaran atau disebut juga dengan negasi atau penyangkalan.
Ingkaran atau Negasi atau Penyangkalan (~)
Dari sebuah pernyataan, kita dapat membuat pernyataan baru berupa ingkaran atau negasi, yakni penyangkalan atas pernyataan tadi. Untuk lebih memahami hal ini, perhatikan tabel kebenaran ingkaran berikut:
Keterangan:
B = pernyataan bernilai benar
S = pernyataan bernilai salah
Artinya, jika suatu pertanyaan (p) benar, maka ingkaran (q) akan bernilai salah. Begitu pula sebaliknya. Nah, negasi ini dilambangkan dengan lambang garis seperti ini: ~
Contoh negasi dalam matematika yaitu seperti berikut:
- p: Besi memuai jika dipanaskan (pernyataan bernilai benar)
- ~p: Besi tidak memuai jika dipanaskan (pernyataan bernilai salah).
Contoh lain:
- p: Semua unggas adalah burung.
- ~p: Ada unggas yang bukan burung.
Dalam kehidupan sehari-hari, kita seringkali menemui orang menggunakan pernyataan negasi atas pernyataan orang lain… yang akhirnya bisa berujung pada pertengkaran. Contohnya seperti gambar di bawah ini, nih!
Baca Juga: Konsep Limit Fungsi Aljabar dan Sifat-Sifatnya
Oke, kembali fokus. Sudah mengerti tentang ingkaran atau negasi, kan? Selanjutnya, kita akan mempelajari tentang pernyataan majemuk. Apa itu pernyataan majemuk?
Pernyataan Majemuk
Dalam ilmu matematika, terdapat 4 macam pernyataan majemuk, yaitu konjungsi, disjungsi, implikasi, dan biimplikasi. Yuk, kita bahas satu per satu!
Konjungsi (∧)
Konjungsi adalah pernyataan majemuk dengan kata hubung “dan”. Sehingga, notasi “p ∧ q” dibaca “p dan q”. Berikut adalah tabel nilai kebenaran konjungsi.
Dari tabel di atas, kita dapat melihat bahwa konjungsi hanya akan benar jika kedua pernyataan (p dan q) benar.
Contoh konjungsi:
- p: 3 adalah bilangan prima (pernyataan bernilai benar)
- q: 3 adalah bilangan ganjil (pernyataan bernilai benar)
- p ∧ q: 3 adalah bilangan prima dan ganjil (pernyataan bernilai benar)
—
Sampai sini, mulai paham kan tentang materi Logika Matematika yang satu ini? Atau kamu jadi keinget punya PR yang kamu masih kurang pahamin? Gampang, kamu bisa banget langsung kirim foto soal PR kamu, dan penjelasannya di roboguru! Cobain langsung dengan klik banner roboguru dibawah ini ya!
Disjungsi (∨)
Disjungsi adalah pernyataan majemuk dengan kata hubung “atau”. Sehingga notasi “p ∨ q” dibaca “p atau q”. Berikut adalah tabel nilai kebenaran disjungsi.
Jika kita lihat pada tabel kebenaran, disjungsi hanya salah jika kedua pernyataan (p dan q) salah.
Contoh disjungsi:
- p: Paus adalah mamalia (pernyataan bernilai benar)
- q: Paus adalah herbivora (pernyataan bernilai salah)
- p ∨ q: Paus adalah mamalia atau herbivora (pernyataan bernilai benar)
Implikasi (⇒)
Implikasi adalah pernyataan majemuk dengan kata hubung “jika… maka…” Sehingga notasi dari “p ⇒ q” dibaca “Jika p, maka q”. Adapun tabel nilai kebenaran dari implikasi yaitu sebagai berikut.
Dari tabel terlihat bahwa implikasi hanya bernilai salah jika anteseden (p) benar, dan konsekuen (q) salah.
Baca Juga: 4 Metode Pembuktian Matematika
Contoh implikasi:
- p: Andi belajar dengan aplikasi ruangguru. (pernyataan bernilai benar)
- q: Andi dapat belajar di mana saja. (pernyataan bernilai benar)
- p ⇒ q: Jika Andi belajar dengan aplikasi ruangguru, maka Andi dapat belajar dari mana saja (pernyataan bernilai benar)
Biimplikasi (⇔)
Biimplikasi adalah pernyataan majemuk dengan kata hubung “… jika dan hanya jika”. Sehingga, notasi dari “p ⇔ q” akan dibaca “p jika dan hanya jika q”. Adapun tabel nilai kebenaran dari biimplikasi yaitu sebagai berikut.
Dari tabel kebenaran tersebut, dapat kita amati bahwa biimplikasi akan bernilai benar jika sebab dan akibatnya (pernyataan p dan q) bernilai sama. Baik itu sama-sama benar, atau sama-sama salah.
Contoh biimplikasi:
- p: 30 x 2 = 60 (pernyataan bernilai benar)
- q: 60 adalah bilangan ganjil (pernyataan bernilai salah)
- p ⇔ q: 30 x 2 = 60 jika dan hanya jika 60 adalah bilangan ganjil (pernyataan bernilai salah).
—
Nah, itu tadi penjelasan tentang logika matematika, baik dalam penggunaan pernyataan dan kalimat terbuka, ingkaran, serta 4 macam kalimat majemuk (konjungsi, disjungsi, implikasi, dan biimplikasi). Kalau kamu ingin memahami materi seperti ini sambil menonton video penjelasan beranimasi lengkap dengan rangkuman infografis dan latihan soal, langsung aja daftar di ruangbelajar!
Referensi:
Sharma S. N., Widiastuti N., Himawan C., dkk. (2017). Jelajah Matematika SMA Kelas XI Program Wajib. Jakarta: Yudisthira.
Artikel ini telah diperbarui pada 9 Mei 2023.
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<h1 class="content-title">
Logika Matematika: Ingkaran, Konjungsi, Disjungsi, Implikasi & Biimplikasi | Matematika Kelas 11 </h1>
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<p class="content-author"><a href="https://www.ruangguru.com/blog/author/kresnoadi" class="">Kresnoadi</a> </p>
<p class="content-info"><span>May 9, 2023 • </span> <span>5 minutes read</span></p>
<br />
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<p><img decoding="async" class="aligncenter" style="width: 820px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/logika%20matematika.jpg" alt="logika matematika" width="820" /></p>
<blockquote>
<p style="text-align: center;"><em><span style="color: #000000;">Belajar tentang <strong>logika matematika</strong>, yuk! Mulai dari pengertian kalimat terbuka, pernyataan dan negasi, serta pernyataan majemuk (konjungsi, disjungsi, implikasi, dan biimplikasi).</span></em></p>
<p style="text-align: center;"><em><span style="color: #000000;">—</span></em></p>
<p> </p></blockquote>
<p style="text-align: justify;"><span style="color: #000000;">Teman-teman, apa yang kamu bayangkan ketika mendengar tentang <a href="/blog/mengulik-materi-logika-matematika" rel="noopener"><strong>logika matematika</strong></a>? Kalau kamu murid laki-laki, mungkin akan bingung dan bertanya, “<em>Kok</em> matematika pakai logika segala?”. Sementara sebagian perempuan akan berpikir, “Logika itu mah urusan laki-laki! Perempuan <em>tuh</em> <em>pake</em>nya perasaan…” </span><span style="color: #000000;">Hmmm…</span></p>
<p style="text-align: justify;"><span style="color: #000000;">Di dalam ilmu matematika, kamu juga dapat mempelajari logika. Buat apa? Tentu aja, buat mengasah otak kita dalam penarikan kesimpulan-kesimpulan. Jadi, ke depannya kita tidak asal menduga sesuatu. Tidak ada lagi deh, kalimat “Kamu bilangnya mau jemput jam 10. <em>Kok </em>telat? Pasti JALAN SAMA MANTAN YA?!”</span></p>
<p>Nah, dalam materi logika matematika, kita akan sering menemukan istilah-istilah, seperti negasi, konjungsi, disjungsi, dan lain sebagainya. Di <strong><a href="https://www.ruangguru.com/blog/tag/matematika-xi" rel="noopener">artikel Matematika kelas 11</a></strong> kali ini, kita akan bahas secara mudah dan ringkas, ya. Yuk, perhatikan secara seksama!</p>
<p> </p>
<h2>Pernyataan dan Kalimat Terbuka</h2>
<p><span style="color: #000000;">Coba kamu perhatikan gambar berikut!</span></p>
<p><img decoding="async" class="aligncenter" style="width: 600px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/pernyataan%20vs%20kalimat%20terbuka.jpg" alt="pernyataan vs kalimat terbuka" width="600" /></p>
<p> </p>
<p>Hayo, dari gambar di atas, tahu nggak bedanya <strong>pernyataan dan kalimat terbuka</strong>? <strong>Pernyataan adalah kalimat yang bisa benar atau bisa salah</strong>. Sementara <strong>kalimat terbuka adalah jenis kalimat yang belum diketahui kebenarannya</strong>. Sehingga, untuk menentukan benar atau salahnya, kita perlu pengamatan lebih lanjut.</p>
<p style="text-align: justify; font-weight: bold;"><span style="color: #000000;">Baca Juga: <a href="/blog/cara-mencari-determinan-dan-invers-matriks" rel="noopener">Cara Mencari Determinan dan Invers Matriks</a></span></p>
<p style="text-align: center;">—</p>
<p style="text-align: center;"><em>Sudah tahu belum, di Aplikasi belajar Ruangguru, ada fitur Drill Soal yang berisi kumpulan contoh soal latihan beserta pembahasannya, loh. Pas banget kan buat mempersiapkan diri kamu dalam menghadapi ujian nanti. Yuk, klik banner di bawah ini untuk coba fitur Drill Soal! </em></p>
<p><a href="https://link.ruangguru.com/blPk/drillsoal15"><img decoding="async" class="aligncenter size-medium" src="https://roboguru-forum-cdn.ruangguru.com/image/c2c1718a-346a-471a-b761-b0a50adcfa50.png" alt="Fitur Drill Soal Ruangguru" /></a></p>
<p> </p>
<p style="text-align: justify;"><span style="color: #000000;">Kalau kamu masih bingung seperti apa itu <strong>contoh pernyataan</strong>, berikut adalah salah satu contohnya:</span></p>
<ul style="text-align: justify;">
<li><span style="color: #000000;">Indonesia Raya adalah lagu kebangsaan Indonesia. (pernyataan <strong>benar</strong>)</span></li>
<li><span style="color: #000000;">Bika ambon berasal dari Ambon. (pernyataan <strong>salah</strong>)</span></li>
</ul>
<p style="text-align: justify;"><span style="color: #000000;">Di sisi lain, <strong>contoh dari kalimat terbuka</strong> adalah sebagai berikut:</span></p>
<ul style="text-align: justify;">
<li><span style="color: #000000;">12x + 6 = 91 (pernyataan ini dinamakan kalimat terbuka karena <strong>masih harus dibuktikan kebenarannya</strong>. Apakah benar 12x jika dijumlahkan dengan 6 akan menghasilkan 91?)</span></li>
</ul>
<ul style="text-align: justify;">
<li><span style="color: #000000;">Maaf ya, aku semalem ketiduran. Hehehe. (Pernyataan ini dinamakan kalimat terbuka karena <strong>masih harus dibuktikan kebenarannya</strong>. Apakah benar dia semalem nggak bales <em>chat</em> karena ketiduran? Atau emang males aja <em>chat </em>sama kamu?)</span></li>
</ul>
<p style="text-align: justify;"><span style="color: #000000;">Nah, setelah mengetahui apa itu pernyataan dan kalimat terbuka, sekarang kita lanjut pembahasan mengenai ingkaran atau disebut juga dengan negasi atau penyangkalan.</span></p>
<p> </p>
<h2>Ingkaran atau Negasi atau Penyangkalan (~)</h2>
<p style="text-align: justify;"><span style="color: #000000;">Dari sebuah pernyataan, kita dapat membuat pernyataan baru berupa <strong>ingkaran atau negasi</strong>, yakni penyangkalan atas pernyataan tadi. Untuk lebih memahami hal ini, perhatikan <span style="font-weight: bold;">tabel kebenaran ingkaran </span><span style="font-weight: normal;">berikut:</span></span></p>
<p style="text-align: justify;"><span style="color: #000000;"><img decoding="async" style="width: 128px;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/tabel%20logika%20matematika.png" alt="tabel logika matematika" width="128" /></span></p>
<p style="text-align: justify; font-weight: bold;"><span style="color: #000000;">Keterangan:</span></p>
<p style="text-align: justify;"><span style="color: #000000;">B = pernyataan bernilai benar</span></p>
<p><span style="color: #000000;">S = pernyataan bernilai salah</span></p>
<p style="text-align: justify;"><span style="color: #000000;">Artinya, <span style="font-weight: bold;">jika suatu pertanyaan (p) benar, maka ingkaran (q) akan bernilai salah</span>. Begitu pula sebaliknya. Nah, negasi ini dilambangkan dengan lambang garis seperti ini: <span style="font-weight: bold;">~</span></span></p>
<p style="text-align: justify;"><span style="color: #000000;"><strong>Contoh negasi dalam matematika</strong> yaitu seperti berikut:</span></p>
<ul>
<li><span style="color: #000000;">p: Besi memuai jika dipanaskan (pernyataan bernilai <strong>benar</strong>)</span></li>
<li><span style="color: #000000;">~p: Besi <strong>tidak</strong> memuai jika dipanaskan (pernyataan bernilai <strong>salah</strong>).</span></li>
</ul>
<p><span style="color: #000000;">Contoh lain:</span></p>
<ul>
<li><span style="color: #000000;">p: <strong>Semua</strong> unggas <strong>adalah</strong> burung.</span></li>
<li><span style="color: #000000;">~p: <strong>Ada</strong> unggas yang <strong>bukan</strong> burung.</span></li>
</ul>
<p style="text-align: justify;"><span style="color: #000000;">Dalam kehidupan sehari-hari, kita seringkali menemui orang menggunakan pernyataan negasi atas pernyataan orang lain… yang akhirnya bisa berujung pada pertengkaran. Contohnya seperti gambar di bawah ini, nih!</span></p>
<p style="text-align: justify;"><span style="background-color: transparent; color: #000000; font-weight: bold;"><img decoding="async" style="width: 600px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/contoh%20kalimat%20negasi.gif" alt="contoh kalimat negasi" width="600" /></span></p>
<p> </p>
<p style="text-align: justify;"><span style="background-color: transparent; color: #000000; font-weight: bold;">Baca Juga: </span><a style="background-color: transparent; font-weight: bold;" href="/blog/konsep-limit-fungsi-aljabar-dan-sifat-sifatnya" rel="noopener">Konsep Limit Fungsi Aljabar dan Sifat-Sifatnya</a></p>
<p style="text-align: justify;"><span style="color: #000000;">Oke, kembali fokus. Sudah mengerti tentang ingkaran atau negasi, kan? Selanjutnya, kita akan mempelajari tentang pernyataan majemuk. <strong>Apa itu pernyataan majemuk</strong>?</span></p>
<p><img decoding="async" class="aligncenter" style="width: 600px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/pernyataan%20majemuk%20dalam%20logika%20matematika.jpg" alt="pernyataan majemuk dalam logika matematika" width="600" /></p>
<p> </p>
<h2>Pernyataan Majemuk</h2>
<p><span style="color: #000000;">Dalam ilmu matematika, terdapat <span style="font-weight: bold;">4 macam</span> pernyataan majemuk, yaitu <span style="font-weight: bold;">konjungsi</span>, <span style="font-weight: bold;">disjungsi</span>, <span style="font-weight: bold;">implikasi</span>, dan <span style="font-weight: bold;">biimplikasi</span>. Yuk, kita bahas satu per satu!</span></p>
<h3>Konjungsi (∧)</h3>
<p><span style="color: #000000;">Konjungsi adalah <strong>pernyataan majemuk dengan kata hubung</strong> “<span style="font-weight: bold;">dan</span>”. Sehingga, notasi “<span style="font-weight: bold;">p ∧ q</span>” dibaca “<span style="font-weight: bold;">p dan q</span>”. Berikut adalah </span><span style="color: #000000;"><span style="font-weight: bold;">tabel nilai kebenaran k</span><strong>onjungsi</strong><span style="font-weight: normal;">.</span></span></p>
<p><img decoding="async" class="aligncenter" style="width: 600px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/tabel%20kebenaran%20konjungsi.jpg" alt="tabel kebenaran konjungsi" width="600" /></p>
<p> </p>
<p><span style="color: #000000;">Dari tabel di atas, kita dapat melihat bahwa <strong>konjungsi hanya akan benar</strong> <strong>jika kedua pernyataan (p dan q) benar</strong><span style="font-weight: normal;">.</span></span></p>
<p style="text-align: justify;"><span style="color: #000000;"><strong>Contoh konjungsi</strong>:</span></p>
<ul>
<li style="text-align: justify;"><span style="color: #000000;">p: 3 adalah bilangan prima (pernyataan bernilai benar)</span></li>
<li style="text-align: justify;"><span style="color: #000000;">q: 3 adalah bilangan ganjil (pernyataan bernilai benar)</span></li>
<li style="text-align: justify;"><span style="color: #000000;">p ∧ q: 3 adalah bilangan prima <strong>dan</strong> ganjil (pernyataan bernilai benar)</span></li>
</ul>
<p style="text-align: center;">—</p>
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<p> </p>
<h3>Disjungsi (∨)</h3>
<p style="text-align: justify;"><span style="color: #000000;"><span style="font-size: 16px;">Disjungsi adalah <strong>perny</strong></span><strong>ataan majemuk dengan kata hubung “atau”</strong>. Sehingga notasi “<span style="font-weight: bold;">p ∨ q</span>” dibaca “<span style="font-weight: bold;">p atau q</span>”. Berikut adalah <span style="font-weight: bold;">t</span></span><span style="color: #000000;"><span style="font-weight: bold;">abel nilai kebenaran disjungsi</span><span style="font-weight: normal;">.</span></span></p>
<p><img decoding="async" class="aligncenter" style="width: 600px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/tabel%20kebenaran%20disjungsi.jpg" alt="tabel kebenaran disjungsi" width="600" /></p>
<p> </p>
<p><span style="color: #000000;"><span style="background-color: transparent;">Jika kita lihat pada tabel kebenaran, disjungsi </span><strong style="background-color: transparent;">hanya salah jika kedua pernyataan (p dan q) salah</strong><span style="font-weight: normal;">.</span></span></p>
<p><span style="color: #000000;"><strong>Contoh disjungsi</strong>:</span></p>
<ul>
<li><span style="color: #000000;">p: Paus adalah mamalia (pernyataan bernilai benar)</span></li>
<li><span style="color: #000000;">q: Paus adalah herbivora (pernyataan bernilai salah)</span></li>
<li><span style="color: #000000;">p ∨ q: Paus adalah mamalia <strong>atau</strong> herbivora (pernyataan bernilai benar)</span></li>
</ul>
<p> </p>
<h3>Implikasi (⇒)</h3>
<p><span style="color: #000000;">Implikasi adalah <strong>pernyataan majemuk dengan kata hubung</strong> “<span style="font-weight: bold;">jika… maka…</span>” Sehingga notasi dari “<span style="font-weight: bold;">p ⇒ q</span>” dibaca “<span style="font-weight: bold;">Jika p, maka q</span>”. Adapun <span style="font-weight: bold;">tabel nilai kebenaran dari implikasi </span><span style="font-weight: normal;">yaitu sebagai berikut.</span></span></p>
<p><img decoding="async" class="aligncenter" style="width: 600px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/tabel%20kebenaran%20implikasi.jpg" alt="tabel kebenaran implikasi" width="600" /></p>
<p> </p>
<p><span style="color: #000000;">Dari tabel terlihat bahwa implikasi <strong>hanya bernilai salah jika anteseden (p) benar, dan konsekuen (q) salah</strong><span style="font-weight: normal;">.</span></span></p>
<p style="font-weight: bold;"><span style="color: #000000;">Baca Juga: <a href="/blog/matematika-kelas-11-pembuktian-matematika" rel="noopener">4 Metode Pembuktian Matematika</a></span></p>
<p><span style="color: #000000;"><strong>Contoh implikasi</strong>:</span></p>
<ul>
<li><span style="color: #000000;">p: Andi belajar dengan aplikasi ruangguru. (pernyataan bernilai benar)</span></li>
<li><span style="color: #000000;">q: Andi dapat belajar di mana saja. (pernyataan bernilai benar)</span></li>
<li><span style="color: #000000;">p ⇒ q: <strong>Jika</strong> Andi belajar dengan aplikasi ruangguru<span style="font-weight: normal;">,</span> <strong>maka</strong> Andi dapat belajar dari mana saja (pernyataan bernilai benar)</span></li>
</ul>
<p> </p>
<h3>Biimplikasi (⇔)</h3>
<p><span style="color: #000000;">Biimplikasi adalah <strong>pernyataan majemuk dengan kata hubung “… jika dan hanya jika”</strong>. Sehingga, notasi dari “<span style="font-weight: bold;">p ⇔ q</span>” akan dibaca “<span style="font-weight: bold;">p jika dan hanya jika q</span>”. </span><span style="color: #000000;">Adapun <span style="font-weight: bold;">tabel nilai kebenaran dari biimplikasi </span><span style="font-weight: normal;">yaitu sebagai berikut.</span></span></p>
<p><img decoding="async" class="aligncenter" style="width: 600px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/tabel%20kebenaran%20biimplikasi.jpg" alt="tabel kebenaran biimplikasi" width="600" /></p>
<p> </p>
<p><span style="color: #000000;">Dari tabel kebenaran tersebut, dapat kita amati bahwa <strong>biimplikasi akan bernilai benar jika sebab dan akibatnya (pernyataan p dan q) bernilai sama</strong><span style="font-weight: normal;">.</span> Baik itu sama-sama benar, atau sama-sama salah.</span></p>
<p><span style="color: #000000;"><strong>Contoh biimplikasi</strong>:</span></p>
<ul>
<li><span style="color: #000000;">p: 30 x 2 = 60 (pernyataan bernilai benar)</span></li>
<li><span style="color: #000000;">q: 60 adalah bilangan ganjil (pernyataan bernilai salah)</span></li>
<li><span style="color: #000000;">p ⇔ q: 30 x 2 = 60 <strong>jika dan hanya jika </strong>60 adalah bilangan ganjil (pernyataan bernilai salah).</span></li>
</ul>
<p style="font-weight: normal; text-align: center;">—</p>
<p style="text-align: justify;"><span style="color: #000000;"><span style="font-style: normal;">Nah</span>, itu tadi penjelasan tentang <span style="font-weight: normal;"><strong>logika matematika</strong>, baik dalam penggunaan pernyataan dan kalimat terbuka, ingkaran, serta 4 macam kalimat majemuk (konjungsi, disjungsi, implikasi, dan biimplikasi). </span>Kalau kamu ingin memahami materi seperti ini sambil menonton video penjelasan beranimasi lengkap dengan rangkuman infografis dan latihan soal, langsung aja daftar di <a href="https://ruangguru.onelink.me/blPk/eaff0eb9" target="_blank" rel="noopener"><strong>ruangbelajar</strong></a><span style="font-weight: normal;">!</span></span></p>
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<p><span style="color: #000000;"><strong>Referensi:</strong></span></p>
<p><span style="color: #000000;">Sharma S. N., Widiastuti N., Himawan C., dkk. (2017). <em>Jelajah Matematika SMA Kelas XI Program Wajib</em>. Jakarta: Yudisthira.</span></p>
<p> </p>
<p style="font-weight: bold;"><span style="color: #000000;"><em>Artikel ini telah diperbarui pada 9 Mei 2023.</em></span></p>
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<li><a href="https://skillacademy.com/" target="_blank">Skill Academy</a></li>
<li><a href="https://www.ruangkerja.id/" target="_blank">Ruangkerja</a></li>
<li><a href="https://www.schoters.com/id/" target="_blank">Schoters</a></li>
<li><a href="https://www.kalananti.id/" target="_blank">Kalananti</a></li>
</ul>
</nav>
</div>
<div class="col-6 col-md-3">
<nav class="footer-menu">
<p class="title"> Coba GRATIS Aplikasi Ruangguru</p>
<div class="download-list">
<ul>
<li>
<a target="_blank" rel="nofollow" href="https://apps.apple.com/id/app/ruangguru-solusi-belajar/id1099742206?l=id">
<img src="https://cdn-web-2.ruangguru.com/static/appstore.png" loading="lazy" alt="Button App Store Footer " />
</a>
</li>
<li>
<a target="_blank" rel="nofollow" href="https://play.google.com/store/apps/details?id=com.ruangguru.livestudents&hl=in">
<img src="https://cdn-web-2.ruangguru.com/static/googleplay.png" loading="lazy" alt="Button Google Play Footer " />
</a>
</li>
<li>
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<img src="https://cdn-web-2.ruangguru.com/static/huawei.png" loading="lazy" alt="Button AppGallery Footer " />
</a>
</li>
</ul>
</div>
</nav>
</div>
</div>
</div>
</div>
<div class="d-lg-none text-center">
<nav class="footer-menu social">
<p class="title">Ikuti Kami</p>
<a target="_blank" href="https://www.instagram.com/ruangguru/">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/instagram.png?height=60&convert=webp" type="image/webp">
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<img src="https://cdn-web-2.ruangguru.com/static/instagram.png" loading="lazy" alt="Logo Instagram Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://www.facebook.com/ruanggurucom/">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/facebook.png?height=60&convert=webp" type="image/webp">
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<img src="https://cdn-web-2.ruangguru.com/static/facebook.png" loading="lazy" alt="Logo Facebook Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://twitter.com/ruangguru/">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/twitter.png?height=60&convert=webp" type="image/webp">
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<img src="https://cdn-web-2.ruangguru.com/static/twitter.png" loading=" lazy" alt="Logo Twitter Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://www.youtube.com/ruangguru">
<picture>
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</picture>
</a>
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</picture>
</a>
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</picture>
</a>
</nav>
</div>
<p class="footer-copyright">
© 2026 All Rights Reserved PT. Ruang Raya Indonesia
</p>
</div>
<style>
.bantuan-container {
position: relative;
}
.bantuan-element {
position: fixed;
z-index: 999;
display: flex;
align-items: flex-end;
justify-content: flex-end;
cursor: pointer;
transition: all 0.6s ease;
}
.bantuan-top {
bottom: 11rem;
right: 2rem;
}
.bantuan-bottom {
bottom: 2rem;
right: 2rem;
}
.bantuan-top-2 {
bottom: 15rem;
right: 2rem;
}
.bantuan-top-3 {
bottom: 19rem;
right: 2rem;
}
.bantuan-bottom-2 {
bottom: 5rem;
right: 2rem;
}
.bantuan-bottom-3 {
bottom: 8rem;
right: 2rem;
}
@media (min-width: 768px) {
.bantuan-element img {
width: auto;
transition: all 0.4s ease;
}
.bantuan-top img {
width: 140px;
}
.bantuan-bottom img {
width: 140px;
}
.bantuan-top-2 img,
.bantuan-top-3 img {
width: 140px;
}
.bantuan-bottom-2 img,
.bantuan-bottom-3 img {
width: 140px;
}
.bantuan-element.clicked {
bottom: 9rem;
right: 2.5rem;
}
.bantuan-element.clicked img {
width: auto;
height: 64px;
}
}
@media (max-width: 767px) {
.bantuan-top {
bottom: 8rem;
right: 1rem;
align-items: center;
}
.bantuan-bottom {
bottom: 2rem;
right: 1.25rem;
align-items: center;
}
.bantuan-top-2 {
bottom: 12rem;
right: 1rem;
align-items: center;
}
.bantuan-top-3 {
bottom: 16rem;
right: 1rem;
align-items: center;
}
.bantuan-bottom-2 {
bottom: 5rem;
right: 1.25rem;
align-items: center;
}
.bantuan-bottom-3 {
bottom: 8rem;
right: 1.25rem;
align-items: center;
}
.bantuan-element img {
width: auto;
}
.bantuan-top img,
.bantuan-top-2 img,
.bantuan-top-3 img {
width: 96px;
}
.bantuan-bottom img,
.bantuan-bottom-2 img,
.bantuan-bottom-3 img {
width: 96px;
}
}
.bantuan-element:hover {
transform: scale(1.05);
}
.bantuan-element img {
max-width: 100%;
height: auto;
}
</style>
<div id="bantuan-divs-container" style="position: relative" data-product="ruangguru" data-table="rg_floating_icon"></div>
<script>
class FloatingDivsComponent {
constructor(product = 'ruangguru', table = 'rg_floating_icon') {
this.container = document.getElementById('bantuan-divs-container');
this.floatingElements = [];
this.product = product;
this.table = table;
// Configuration
this.config = {
baseApiUrl: 'https://gw.ruangguru.com/api/v3/rdb/p_ddgy8hjx5dxsqx/rg_banner/views/',
apiKey: 'vLyK3dJrDBRmOwueX2OJXibMkgsUbBk0WS4ua2p1',
maxElements: 2,
zIndexBase: 100,
};
// Build dynamic API URL based on table parameter
this.config.apiUrl = this.buildApiUrl();
this.init();
}
buildApiUrl() {
const baseUrl = this.config.baseApiUrl;
const tableName = this.table || 'rg_floating_icon';
const queryParams =
'offset=0&limit=99999&sort=rankPosition&where=(type,eq,floating)~and(showIn,anyof,blog)~and(hide,is,false)~and(product,eq,' +
this.product +
')&nested%5Bcta%5D%5Bfields%5D=Id,name,ctaId,imageUrl,ctaLink,targetUrl';
return `${baseUrl}${tableName}?${queryParams}`;
}
async init() {
try {
await this.fetchFloatingElements();
this.renderFloatingElements();
} catch (error) {
console.error('Failed to initialize floating divs component:', error);
}
}
async fetchFloatingElements() {
try {
const options = {
method: 'GET',
headers: {
'xc-token': this.config.apiKey || 'vLyK3dJrDBRmOwueX2OJXibMkgsUbBk0WS4ua2p1',
},
};
const urlNoco = this.config.apiUrl;
const response = await fetch(urlNoco, options);
if (!response.ok) {
throw new Error(`HTTP error! status: ${response.status}`);
}
const data = await response.json();
this.floatingElements = this.processApiResponse(data);
} catch (error) {
console.error('Error fetching floating elements:', error);
throw error;
}
}
processApiResponse(data) {
if (!data || !data.list || !Array.isArray(data.list)) {
return [];
}
console.log(data.list);
// Ensure we only get maximum 2 elements
const limitedElements = data.list.slice(0, this.config.maxElements);
return limitedElements;
}
renderFloatingElements() {
if (!this.container || this.floatingElements.length === 0) {
return;
}
// Additional safeguard: ensure we never render more than maxElements
const elementsToRender = this.floatingElements.slice(0, this.config.maxElements);
// Clean up existing elements and their event listeners
this.cleanupExistingElements();
// Clear existing content
this.container.innerHTML = '';
elementsToRender.forEach((element, index) => {
const floatingDiv = this.createFloatingDiv(element, index);
this.container.appendChild(floatingDiv);
});
}
cleanupExistingElements() {
// Clear existing elements (no need for event listener cleanup with CSS approach)
const existingElements = this.container.querySelectorAll('[data-floating-element]');
existingElements.forEach(element => {
element.remove();
});
}
createFloatingDiv(element, index) {
const div = document.createElement('div');
const cta = element.cta[0]; // Use the first CTA
// Add data attribute for cleanup tracking
div.setAttribute('data-floating-element', 'true');
// Apply CSS classes based on rankPosition
const positionClass = this.getPositionClass(element.rankPosition, index);
div.className = `bantuan-element ${positionClass}`;
div.style.zIndex = this.config.zIndexBase + index;
// Create the link and image
const link = document.createElement('a');
link.href = 'https://cta.ruangguru.com/' + cta.ctaId || '#';
link.target = '_blank';
// Add tracking attributes
link.setAttribute('data-tracking-event-type', 'landing_page__core__item_click');
link.setAttribute(
'data-tracking-context',
JSON.stringify({
product_name: this.product,
cta_action: 'redirection',
cta_action_detail: link.href,
item_type: 'floating',
item_name: element.name || 'Floating element',
item_category: 'payment',
item_position: index,
item_serial: null,
section_type: 'floating_item',
section_position: 'body',
section_name: 'floating_item',
})
);
const img = document.createElement('img');
img.src = element.imageUrl + '?h=200&convert=webp';
img.alt = element.name || 'Floating element';
img.loading = 'lazy';
// Add click tracking
link.addEventListener('click', () => {
this.trackClick(element, cta);
});
link.appendChild(img);
div.appendChild(link);
return div;
}
getPositionClass(rankPosition, index) {
// Define positioning classes based on rankPosition
if (rankPosition === 1) {
return 'bantuan-top'; // Top position
} else if (rankPosition === 2) {
return 'bantuan-bottom'; // Bottom position
} else {
// For additional elements, distribute them evenly
return `bantuan-${index % 2 === 0 ? 'top' : 'bottom'}-${Math.floor(index / 2) + 1}`;
}
}
trackClick(element, cta) {
// Implement click tracking logic here
console.log('Floating element clicked:', {
elementId: element.Id,
elementName: cta.name,
ctaId: cta.ctaId,
targetUrl: cta.targetUrl,
});
}
// Public method to manually refresh
async refresh() {
try {
await this.fetchFloatingElements();
this.renderFloatingElements();
} catch (error) {
console.error('Failed to refresh floating elements:', error);
throw error;
}
}
// Public method to get current floating elements
getFloatingElements() {
return [...this.floatingElements];
}
}
// Initialize the component when DOM is ready
document.addEventListener('DOMContentLoaded', () => {
// Get product and table parameters from data attributes
const container = document.getElementById('bantuan-divs-container');
const product = container.getAttribute('data-product') || 'ruangguru';
const table = container.getAttribute('data-table') || 'rg_floating_icon';
new FloatingDivsComponent(product, table);
});
// Export for use in other modules if needed
if (typeof module !== 'undefined' && module.exports) {
module.exports = FloatingDivsComponent;
}
</script> <div id="wp-install-nudge" style="
position: fixed;
bottom: 0rem;
right: 0;
z-index: 999;
cursor: pointer;
"></div>
<script>
(function (w, d, s, o, f, js, fjs) {
w['Sales-Nudge-Widget'] = o; w[o] = w[o] || function () { (w[o].q = w[o].q || []).push(arguments) };
js = d.createElement(s), fjs = d.getElementsByTagName(s)[0];
js.id = o; js.src = f; js.async = 1; fjs.parentNode.insertBefore(js, fjs);
}(window, document, 'script', 'wpInstallNudge', 'https://cdn-web-2.ruangguru.com/sales-nudge/6198a95130/dist/widget.js'));
wpInstallNudge('init', { targetElementId: 'wp-install-nudge', clientTarget: 'ruangguru_blog' });
</script>
<script>
// note: disabled due to auto reload page
// var lastScrollTop = 0;
// $(window).scroll(function(event){
// var st = $(this).scrollTop();
// if (st > lastScrollTop){
// $('#wp-install-nudge').slideDown( "slow", function() {});
// } else {
// $('#wp-install-nudge').slideUp( "slow", function() {});
// }
// lastScrollTop = st;
// });
// $(document).ready(function () {
// $("#wp-install-nudge").hide();
// });
$('#wp-install-nudge').click(function(){
Analytics.track("promotion__install_nudge__click",
{
user_email: "ruangguru@blog.com",
cta_link: widgetData.dataConfig.target_link
}).catch((err) => {
console.log("tracker error", err);
});
});
</script>
</footer>
<div id="__next"></div>
<script type="speculationrules">
{"prefetch":[{"source":"document","where":{"and":[{"href_matches":"\/blog\/*"},{"not":{"href_matches":["\/blog\/wp-*.php","\/blog\/wp-admin\/*","\/blog\/wp-content\/uploads\/*","\/blog\/wp-content\/*","\/blog\/wp-content\/plugins\/*","\/blog\/wp-content\/themes\/ruangguru\/*","\/blog\/*\\?(.+)"]}},{"not":{"selector_matches":"a[rel~=\"nofollow\"]"}},{"not":{"selector_matches":".no-prefetch, .no-prefetch a"}}]},"eagerness":"conservative"}]}
</script>
<script>
async function getCtaDetails(ctaIds) {
const endpoint =
" https://gw.ruangguru.com/api/v3/rdb/p_ddgy8hjx5dxsqx/CTA/views/CTA?offset=0&limit=10&where=(ctaId,in," + ctaIds.toString() + ")";
try {
const response = await fetch(endpoint, {
headers: {
"xc-token": "-hG2nBDJzSj0agSgNtcJx97M4F02y54nUMoY6plJ",
},
});
const data = await response.json();
const ctaList = data?.list || [];
const ctaMap = new Map();
for (const ctaData of ctaList) {
ctaMap.set(ctaData.ctaId, ctaData);
}
return ctaMap;
} catch (error) {
console.error("Error fetching CTA details:", error);
return new Map();
}
}
async function replaceCTA() {
const regex = /{{cta\((‘(.*?)’|“(.*?)”)\)}}/g;
const bodyHTML = document.body.innerHTML;
const matches = Array.from(bodyHTML.matchAll(regex));
const ctaIds = matches.map((match) => {
let ctaId = match[2] ?? match[3];
return ctaId;
});
const ctaMap = await getCtaDetails(ctaIds);
const replacedHTML = bodyHTML.replace(regex, (match, _inner, ctaId1, ctaId2) => {
const ctaId = ctaId1 ?? ctaId2;
const ctaData = ctaMap.get(ctaId);
if (ctaData) {
return `<a href="https://cta.ruangguru.com/${ctaData.ctaId}" target="_${ctaData.targetType}" style="position:relative;display:inline-block;">
<img src="${ctaData.imageUrl}" loading="lazy" alt="${ctaData.altText}" />
</a>`;
} else {
return match;
}
});
document.body.innerHTML = replacedHTML;
}
</script>
<meta name="author" content="Ruangguru Tech Team">
<!-- Google Tag Manager (noscript) -->
<noscript><iframe src="https://www.googletagmanager.com/ns.html?id=GTM-PVNLZFT" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript>
<!-- Tracker -->
<script type="text/javascript" src="https://cdn-web-2.ruangguru.com/landing-page-web/public/staticpages/global-assets/js/cta.js"></script>
<script src="https://unpkg.com/@dotlottie/player-component@latest/dist/dotlottie-player.mjs" type="module"></script>
</body>
</html>