The Cardinality of a Set:
The cardinality of a set S, denoted by |S|, is the number of elements of the set. The number is also referred to as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞.
Example − |{1, 4, 3, 5}| = 4, |{1, 2, 3, 4, 5,….}| = ∞
If there are two sets X and Y,
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|X| = |Y| denotes two sets X and Y have the same cardinality. It occurs when the number of elements in X is exactly equal to the number of elements in Y.
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|X| ≤ |Y| denotes that set X’s cardinality is less than or equal to set Y’s cardinality. It occurs when the number of elements in X is less than or equal to that of Y.
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|X| < |Y| denotes that set X’s cardinality is less than set Y’s cardinality. It occurs when the number of elements in X is less than that of Y.
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If |X| ≤ |Y| and |X| ≥ |Y| then |X| = |Y|. The sets X and Y are commonly referred to as equivalent sets.