Law-Forums.org

[gedalyahu]

Let X, Y and Z be any sets. Prove the following statements otherwise provide a counter-example.

a) Y ? (Z – X) = Z ? (Y – X)

b) (Y ? Z) – X = (Y – Z) ? X

For a) this is how I did it, not sure if it's right or not:

Y ? (Z – X) = Y ? Z ? Xc since Z – X = Z ? Xc (Xc = compliment of X)
Y ? (Z – X) = Z ? Y ? Xc due to commutative property
Y ? (Z – X) = Z ? (Y – X)

However for b) I'm not very sure how to do it. Could someone help me with this please?

[orson19]

This is because b) is untrue!

Q = Y ? Z and R = Y – Z are disjoint just like Q – X and R ? X are disjoint. Q is a subset of Z whereas R is a subset of Zc. Similarly Q – X is a subset of Xc whereas R ? X is a subset of X. So unless both Z and X are empty sets the statement is untrue.

[curadhan74]

You should prove it rigorously for both (a) and (b). When showing two sets are equal you need to show that each is a subset of the other. So here you need to show
i) Y ? (Z – X) is a subset of Z ? (Y – X)
ii) Z ? (Y – X) is a subset of Y ? (Z – X)

(i) Let x be an element of Y ? (Z – X) then x belongs to Y and x belongs to (Z-X). The latter implies that x does not belong to X. Since x belong to Y and x does not belong to X we must have x belongs to (Y-X). Now x belongs to Z and x belongs to (Y-X), therefore x belongs to the intersection of the two sets. Hence Y ? (Z – X) is a subset of Z ? (Y – X).
You can try doing (ii) yourself.

Now you can't prove (b) since it is not true. Here you need to give a counterexample
For example
Let X = {-1,-2,-3,0,1,2,3}
Let Y = {1,2,4,5,6}
Let Z = {1,4,9}
Then (Y ? Z) – X = {4}
But (Y – Z) ? X = {2}