Let A, B, C, D be sets with a 1-1 correspondence between A and C and another 1-1

Question

Let A, B, C, D be sets with a 1-1 correspondence between A and C and another 1-1 correspondence between B and D. Prove that if A and B are disjoint, and C and D are also disjoint, then there is a 1-1 correspondence between A union B and C union D Let A, B, C, D be sets with a 1-1 correspondence between A and C and another 1-1 correspondence between B and D. Prove that if A and B are disjoint, and C and D are also disjoint, then there is a 1-1 correspondence between A union B and C union D Let A, B, C, D be sets with a 1-1 correspondence between A and C and another 1-1 correspondence between B and D. Prove that if A and B are disjoint, and C and D are also disjoint, then there is a 1-1 correspondence between A union B and C union D

Explanation / Answer

Since A & B are disjoint, A union B is A+B. That is all the terms of A + all the terms of B. Neither elimianted nor repeated. So the number of terms would be (a+b)

Similarly we have number of terms for C union D is c+d.

since A & C have 1-1 correspondence, so a is equal to c

Similarly b is equal to d.

So a+b is equal to c+d

Now the cardinalty of both are same, let's check the elements.

so if A is 1-1 correspondence with C then every element in A is distinctly related to C and is same with B & D.

So since there is no repeat or elimination of terms, therfore A+B is in 1-1 correspondence with C+D

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