- Cartesian Product of Empty Set
- The empty set is a unique set with no elements. Both its size or cardinality i.e, the total count of elements in a set will remain zero. An empty set is also referred to as a void set. The Cartesian product of C and the empty set ∅ is the empty set ∅. Let C × ∅ = {(a,b)| a ∈ C, b∈ ∅}. There is no element in ∅. C × D =∅ if and only if C = ∅ or D = ∅. Here, the cartesian product of two sets will result in an empty set if and only if, either of the sets is an empty set.
- Consider the example: If C = {1, 2} and D = ϕ. Then, C × D = ϕ and D × C = ϕ.
These are the properties of the empty set: - These are the properties of the empty set:
- Empty set's subset is the empty set itself: ∀ C:C ⊆ ∅ ⇒ C = ∅
- The empty set's power set is the set containing only the empty set: 2n = 20 = 1.
- The cardinality of the empty set i.e., the number of elements of the set is zero: n(∅) = 0
- Cartesian Product of Countable Sets
- The cartesian product of two countable sets is countable. Let us take these two cases to understand this:
- Consider an integer b in such a way that b > 1. Then the cartesian product of b countable sets is countable.
- Consider the two countable sets A = {a0, a1, a2, ...} and B = {b0, b1, b2, ...}. If both the sets A and B are countable, then the resulting set will also be coutable.
Cartesian Product Of Relations - The cartesian product of relations is the same as the relation across two sets. Generally, the cartesian product is represented for a set and not for a relation. Further, the universal relation relates every element of one set to an element of another set, and hence it can be represented as the cartesian product of relations.
- Cardinality of a Cartesian Product
- The cardinality of a set is the total number of elements present in the set. The cardinal number of A is n(A) = number of all the elements in set A. Example: The cardinal number of a set of English alphabets A = (a, b, c .....x. y. z) is n(A) = 26. The сardinality of a cartesian product of two sets C and D is equal to the product of the cardinalities of these two sets: n(C × D) = n(D × C) = n(C) × n(D). Similarly, n(C1×…× Cn) = n(C1) ×…× n(Cn).
- Consider two sets C and D, where C = {2,3} and n(C) = 2, D = {5,4,7} and n(D) = 3. So, n(C × D) = n(C) × n(D) = 2 × 3 = 6. Here, we can see that the cardinality of the output set C × D is equal to the product of the cardinalities of all the input sets C and D. That is, 6.
- Important Notes on Cartesian Product
- Sometimes, ordered pairs are also referred to as 2−tuples.
- The cartesian product of two sets C and D is also known as the cross-product or the product set of C and D
- The final cartesian product of two sets will be a collection of all ordered pairs obtained by the product of these two non-empty sets.
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