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Discrete math problems - theorem

In basic algebra the following Theorem is used frequently.
If x,y and z are any three real numbers and if x + z = y + z then x = y.

The analogous statement for sets would read:
Let A, B, and C be any three sets.

If A union B = A union C then B = C.

Prove in detail that this statement is false. (Hint: Give a counterexample)


Write the converse of the above statement and show through an example that it is true.

Please see attached for complete details.

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Solution Summary

Attached in this posting is the solution and the explanations on how the answer is derived.