if A={1,2,3,4,5,6} then
cardionity of set A.
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What is the cardinality of set | {1, 2, 3, 4, 5} ^5| and | {9} ^4|? I don't understand how the exponets work.
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3 Answers
Roger Pickering, Spent 6 years at 2 universities doing maths
Answered November 12
If A is a set, and it can be finite or infinite, then A² is shorthand for the product set AxA.
That is the set of pairs of elements (a,b) where a and b both belong to A.
If A is finite with n elements then AxA has n² elements.
Similarly A³ is the set AxAxA of triples (a,b,c) with all three belonging to A.
So now we come to your question. The first set {1,2,3,4,5} has five elements, so A⁵=AxAxAxAxA and has 5x5x5x5x5=5⁵=3125 elements.
Your second set which I shall call B={9} has only 1 element, so when we make BxB or BxBxBxB, we don't have any choice over the elements (b,c) and (b,c,d,e). They all have to equal 9, and those product sets also only have 1 element. In the case of B⁴=BxBxBxB the only element is (9,9,9,9) and the cardinality of B⁴ is 1.