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If it's true, prove it, if not give specific sets $A,B,C,D.$

My try: I tried to work with some sets and the statement turned out to be true. But when Im using identities, I do not know how to manipulate the RHS

  • for the LHS: $( A \times B ) \cap ( A \times C' ) $
  • but RHS: ?

Any help would be greatly appreciated ! thank you

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  • $\begingroup$ Typesetting hint: you probably want \setminus instead of / so A \setminus B gives $A \setminus B$ $\endgroup$ Commented Jul 22, 2019 at 3:50

1 Answer 1

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We want to prove, that $A\times (B-C)=(A\times B)-(A\times C)$

So let $(x,y)\in A\times (B-C)$. Then $x\in A$ and $y\in B$ and $y\notin C$. Hence $(x,y)\in A\times B$ and $(x,y)\notin A\times C$ and we conclude $(x,y)\in (A\times B)-(A\times C)$.

Go through this proof one by one. If you struggle with one point, go and look at the definition of the symbols.

Now let $(x,y)\in (A\times B)-(A\times C)$. So $(x,y)\in (A\times B)$ and $(x,y)\notin (A\times C)$.

$(x,y)\in A\times B$. Hence $x\in A$ and $y\in B$.

Since $(x,y)\notin A\times C$ and we know, that $x\in A$ we have that $y\notin C$.

So $y\in B-C$ and we conclude $(x,y)\in A\times (B-C)$.

Indeed both sets are equal.

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