If x and y and a be three non-empty sets such that a intersection x equals a intersection y and a union x equals a union y then prove that x equals y?

Answer 1

Let x, y, and a be sets and X,Y,x',y' be elements.

Denote X *x as X in (is an element of) x, I as intersection, and U as union.

If we can show that for all X *x, X *y (and similarly, if for all Y *y, Y *x), then we are done.

Case 1) xIa is empty

Then x, a and y, a have no elements in common. So, if xUa and yUa are equal, then for all y' *yUa but y' not*a, y' *y. Since xUa and yUa are equal, either y' *a or y' *x. But we supposed y' is not*a, so y'*x. Similarly, for all x' *x, x'*y. QED

Case 2) xIa is non-empty

Define a' as a - {x| x *xIa}. Then xIa' is empty, and you can use the same prove as above, replacing a with a'. QED