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Question

Let n(A)=m and n(B)=n. Then, the total number of non-empty relations that can be defined from A to B is .

A
mn
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B
mn1
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C
2mn1
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D
nm1
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E
2mn
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Solution

The correct option is C 2mn1
Let n(A)=m and n(B)=n.
We have A×B={(a,b):aA,bB}
n(A×B)=n(A)×n(B)=mn

A relation from A to B is a subset of A×B.
Since A×B has mn elements, it has 2mn subsets.
Thus, there can be 2mn relations that can be defined from A to B.
The total number of non-empty relations (excluding the subset ϕ of A×B) that can be defined from A to B is 2mn1.

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