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Cardinality of a Set
Cardinality of a set S, denoted by |S|, is the number of elements of the set. The number is also referred 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,
|X| = |Y| denotes two sets X and Y having same cardinality. It occurs when the number of elements in X is exactly equal to the number of elements in Y. In this case, there exists a bijective function ‘f’ from X to Y.
|X| ≤ |Y| denotes that set X’s cardinality is less than or equal to set Y’s cardinality. It occurs when number of elements in X is less than or equal to that of Y. Here, there exists an injective function ‘f’ from X to Y.
|X| < |Y| denotes that set X’s cardinality is less than set Y’s cardinality. It occurs when number of elements in X is less than that of Y. Here, the function ‘f’ from X to Y is injective function but not bijective.
If |X| ≤ |Y| and |X| ≥ |Y| then |X| = |Y|. The sets X and Y are commonly referred as equivalent sets.
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