set distributive proof

[MathsNoobNoob]

Please help to proof the above statement

 

[Dr.Peterson]

You want to show that any element of the set [MATH]A\times B- A\times C[/MATH] is an element of [MATH]A\times(B - C)[/MATH]. Can you describe what any element of the former looks like? It will be an ordered pair (a, b), where ... what?

Then what do you have to show about it?

 

[pka]

View attachment 18246 Please help to proof the above statement

Suppose that \((x,y)\in(A\times B)\setminus (A\times C)\)
That means \([(x,y)\in (A\times B)]\wedge [(x,y)\notin (A\times C)]\).
Which translates: \([x\in A\wedge y\in B]\wedge[x\notin A\vee y\notin C]\)
Proper distribution leaves \([\text{FALSE}]\vee[x\in A\wedge y\in B\wedge y\notin C]\) How and WHY?
But that means \((x,y)\in A\times(B\setminus C)\)
How is that the proof?