0 added
0 removed
Original
2026-01-01
Modified
2026-02-28
1
<p><strong>Empty Relation</strong></p>
1
<p><strong>Empty Relation</strong></p>
2
<p>Definition: In empty relations, no elements are related. Example: For student set A, the relation “is taller than” is empty if all students are the same height: R = or R = {}.</p>
2
<p>Definition: In empty relations, no elements are related. Example: For student set A, the relation “is taller than” is empty if all students are the same height: R = or R = {}.</p>
3
<p><strong>Universal Relation</strong></p>
3
<p><strong>Universal Relation</strong></p>
4
<p>Definition: Every pair of elements is related Example: For set A = {1, 2}, R = {(1, 1), (1, 2), (2, 1), (2, 2)} = A × A</p>
4
<p>Definition: Every pair of elements is related Example: For set A = {1, 2}, R = {(1, 1), (1, 2), (2, 1), (2, 2)} = A × A</p>
5
<p><strong>Identity Relation</strong></p>
5
<p><strong>Identity Relation</strong></p>
6
<p>Definition: A relation in which every element in a set is paired with itself only. Example: For A = {a, b, c}, R = {(a, a), (b, b), (c, c)}</p>
6
<p>Definition: A relation in which every element in a set is paired with itself only. Example: For A = {a, b, c}, R = {(a, a), (b, b), (c, c)}</p>
7
<p><strong>Reflexive Relation</strong></p>
7
<p><strong>Reflexive Relation</strong></p>
8
<p>Definition: Every element in the set is related to itself. It can also include other pairs, but the key requirement is that each element must relate to itself, such as (a, a), (b, b), etc. Example: On A = {1, 2}, R = {(1, 1), (2, 2), (1, 2)} </p>
8
<p>Definition: Every element in the set is related to itself. It can also include other pairs, but the key requirement is that each element must relate to itself, such as (a, a), (b, b), etc. Example: On A = {1, 2}, R = {(1, 1), (2, 2), (1, 2)} </p>
9
<p><strong>Symmetric Relation</strong></p>
9
<p><strong>Symmetric Relation</strong></p>
10
<p>Definition: If a relates to b, then b relates to a Example: R = {(2, 3), (3, 2) on (2, 3)} </p>
10
<p>Definition: If a relates to b, then b relates to a Example: R = {(2, 3), (3, 2) on (2, 3)} </p>
11
<p><strong>Transitive Relation</strong></p>
11
<p><strong>Transitive Relation</strong></p>
12
<p>Definition: A relation is called transitive if, whenever an element a is related to b, and b is related to c, then a must also be related to c. Example: For a set A = {1, 2, 3}, the transitive relation is given as R = {(1, 2), (2, 3), (1, 3)} </p>
12
<p>Definition: A relation is called transitive if, whenever an element a is related to b, and b is related to c, then a must also be related to c. Example: For a set A = {1, 2, 3}, the transitive relation is given as R = {(1, 2), (2, 3), (1, 3)} </p>
13
<p><strong>Equivalence Relation</strong></p>
13
<p><strong>Equivalence Relation</strong></p>
14
<p>Definition: An equivalence relation is reflexive, symmetric, and transitive altogether. Example: On the<a>set of real numbers</a>R, the relation “is equal to” (=) is an equivalence relation because: Every number equals itself (reflexive) If a = b, then b = a (symmetric) If a = b and b = c, then a = c (transitive)</p>
14
<p>Definition: An equivalence relation is reflexive, symmetric, and transitive altogether. Example: On the<a>set of real numbers</a>R, the relation “is equal to” (=) is an equivalence relation because: Every number equals itself (reflexive) If a = b, then b = a (symmetric) If a = b and b = c, then a = c (transitive)</p>
15
<p><strong>Antisymmetric Relation</strong></p>
15
<p><strong>Antisymmetric Relation</strong></p>
16
<p>Definition: A relation is antisymmetric if, whenever (a, b) and (b, a) are both in the relation, then it must be that a = b. Example: Consider the relation "≤" (<a>less than</a>or equal to) on numbers. If a ≤ b and b ≤ a, then the only way both can be true is if a = b. So, "≤" is an antisymmetric relation.</p>
16
<p>Definition: A relation is antisymmetric if, whenever (a, b) and (b, a) are both in the relation, then it must be that a = b. Example: Consider the relation "≤" (<a>less than</a>or equal to) on numbers. If a ≤ b and b ≤ a, then the only way both can be true is if a = b. So, "≤" is an antisymmetric relation.</p>
17
<p><strong>Inverse Relation</strong></p>
17
<p><strong>Inverse Relation</strong></p>
18
<p>Definition: The inverse of a relation is formed by reversing the order of each pair in the original relation. Example: If R = {(1, 2), (3, 5)}, then the<a>inverse relation</a>R-1 = {(2, 1), (5, 3)} </p>
18
<p>Definition: The inverse of a relation is formed by reversing the order of each pair in the original relation. Example: If R = {(1, 2), (3, 5)}, then the<a>inverse relation</a>R-1 = {(2, 1), (5, 3)} </p>
19
19