Rivalry2

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Original 2026-01-01

Modified 2026-02-28

1 Sets can be divided into various types according to their attributes, based on the elements they contain or the relationship with other sets. Singleton sets, finite and<a>infinite sets</a>, equal and unequal sets, equivalent sets, overlapping and<a>disjoint sets</a>, subsets, supersets,<a>power</a>sets, and universal sets.

2 Singleton sets: A<a>singleton set</a>is a set that has a single element. It is called a unit set because it contains only one element.

3 For example, the only<a>number</a>in the set is 6: \(S = \{6\}\)

4 Properties of a singleton set:

5 <ol><li>There is only one element in a singleton set. For example, {6}. </li>

6 <li>Its size, or dimension, is 1. Because it has only one element. </li>

7 <li>A finite set is always a singleton set.</li>

8 </ol>Finite sets: A finite set contains a finite or exactly countable number of elements.

9 For example, the set \(\{20, 40, 60, 80, 100\}\) contains<a>even numbers</a>. This set has 5 countable elements.

10 \(S = \{20, 40, 60, 80, 100\}\)

11 Properties of a finite set:

12 <ol><li>The set is countable because a finite set has a fixed number of elements. </li>

13 <li>With zero elements, the<a>empty set</a>is also finite. </li>

14 <li>Every singleton set is a finite set, but not all finite sets are singletons.</li>

15 </ol>Infinite sets: An infinite set contains an infinite number of elements.

16 For example, prime numbers are greater than 1, and they have only two factors: 1 and themselves.

17 \(D = \{2, 3, 5, 7, 11, 13, 17, 19,...\}\)

18 The list of prime numbers has no end.

19 Properties of an infinite set

20 <ol><li>It contains many elements that cannot be counted. </li>

21 <li>It is impossible to count every element; it is uncountable. </li>

22 <li>Adding or removing elements does not make a set finite or infinite.</li>

23 </ol>Equal sets: Two sets are equal if they contain the same elements, regardless of what order the elements are in.

24 For example: \(L = \{blue, violet, orange\}\) and \(D = \{orange, violet, blue\}\) are equal because they contain the same elements.

25 Properties of equal sets

26 <ol><li>If every element in one set is also present in the other, then the two sets are equal. </li>

27 <li>In sets, the order of elements can be irrelevant. </li>

28 <li>Two sets are equal when they contain the same elements, even if the order changes.</li>

29 </ol>Unequal sets: Two sets are considered unequal if there is at least one element that differs between them.

30 Example: Let \(B = \{pineapple, banana, apple\}\) and \(P = \{pineapple, banana, guava\}\). Here, set B and set P are unequal sets because even if one element is different, the sets become unequal.

31 Properties of unequal sets:

32 <ol><li>The sets differ, even though one element is not the same.</li>

33 <li>Sets might or might not contain the same number of elements.</li>

34 </ol>Equivalent sets: When two sets have the same number of elements, even when they are not the same, they are referred to as equivalent sets.

35 Example: Set \(O = \{2, 5, 6, 7\}\) and \(L = \{w, x, y, z\}\). Since \(n(O) = n(L)\), sets O and L are equivalent in this case.

36 Properties of equivalent sets:

37 <ol><li>It has the same number of elements. </li>

38 <li>The number of elements is the same, but the elements themselves differ from each other. </li>

39 <li>Not all equivalent sets are equal, but all equal sets are equivalent.</li>

40 </ol>Overlapping sets: If at least one item from set A appears in set B, then the two sets are said to overlap.

41 Example: Set \(X = \{2, 4, 6\}\) and \(Y = \{5, 10, 2\}\). In this case, element 2 is present in both sets X and Y. So, they are said to overlap or intersect.

42 Properties of overlapping sets:

43 <ol><li>Disjoint sets do not have common elements, and overlapping sets share some elements. </li>

44 <li>If two sets have common elements, they are also known as intersecting sets. \(A = \{1, 3, 5, 7, 9\}\) </li>

45 </ol>Disjoint sets: If two sets do not have common elements, they are said to be disjoint sets.

46 Example: Let us assume, set \(U = \{1, 3, 5, 7\}\) and \(S = \{2, 4, 8, 6\}\). The sets U and S are disjoint in this case.

47 Properties of disjoint sets:

48 <ol><li>The empty set is the intersection of sets because they don’t have common elements in a disjoint set.

49 </li>

50 <li>Both finite and infinite disjoint sets are possible.

51 </li>

52 <li>Even if one or both of the sets are empty, they can still be disjoint.

53 </li>

54 </ol>Subset and superset: If every element in set A is also present in Set B, then set A is a subset of set B. If set \((A ⊆ B)\), then set B is the superset of set A \((B ⊇ A)\).

55 Example: Let \(A = \{10, 50, 30\}\), \(B = \{10, 50, 30, 7,1, 2\}\).

56 \(A ⊆ B\), since every element in set A is present in set B, \(B ⊇ A\) indicates that set B is set A’s superset.

57 Properties of subset and superset

58 <ol><li>Every set is a subset of itself. </li>

59 <li>Supersets can be larger than another set or equal to it.</li>

60 </ol>Universal sets: The collection of all the elements related to a particular topic is known as a universal set.

61 Example: Let \(P = \{The\ list \ of\ all\ airline \ vehicles\}\) that contains helicopters, jets, and rockets. In this case, if P is the set of all airline vehicles, then jets, helicopters, and rockets are subsets of P because they are the types of airline vehicles.

62 Properties of universal sets:

63 <ol><li>A universal set can be finite or infinite, depending on the context. </li>

64 <li>All sets are subsets of the universal set. </li>

65 <li>To find a set’s complement, subtract its elements from the universal set. </li>

66 </ol>Power sets: A power set is the collection of all possible subsets within a set.

67 Example: All possible subsets of A are included in the power set of \(A = \{m, n\}\), which is \(P(A) = \{∅, \{m\}, \{n\}, \{m, n\}\}\).

68 Properties of power sets:

69 <ol><li>The power set of set A contains all possible subsets of A. </li>

70 <li>A set with n elements will have 2n subsets in its power set. </li>

71 <li>The power set always has more elements than the original set if the original set is not empty.</li>

72 </ol>