Question

Let A nd B bw sets . show that f : A×B - B×A Such that f (a,b) = (b,a) is bijective function

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Answer 1

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Answer 2

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To show that the function f: A×B -> B×A defined as f(a,b) = (b,a) is a bijective function, we need to prove two properties: injectivity (one-to-one) and surjectivity (onto).

1. Injectivity:

Let (a1, b1) and (a2, b2) be two distinct elements in A×B. We need to show that if f(a1, b1) = f(a2, b2), then (a1, b1) = (a2, b2).

f(a1, b1) = (b1, a1)

f(a2, b2) = (b2, a2)

If (b1, a1) = (b2, a2), it implies b1 = b2 and a1 = a2. Therefore, (a1, b1) = (a2, b2).

Hence, the function f is injective.

2. Surjectivity:

For every element (b, a) in B×A, we need to find an element (a', b') in A×B such that f(a', b') = (b, a).

Let a' = a and b' = b. Now f(a', b') = f(a, b) = (b, a).

Therefore, for every element (b, a) in B×A, we can find an element (a', b') in A×B such that f(a', b') = (b, a).

Hence, the function f is surjective.

Since f is both injective and surjective, it is bijective.

Therefore, the function f: A×B -> B×A, defined as f(a,b) = (b,a), is a bijective function.

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