Prove that A X (B U C) = (A X B) U (A X C)

Prove that A X (B U C) = (A X B) U (A X C)

In English this is saying to prove that the Cartesian product of A and B union C is equal to the Cartesian product of A and B union the Cartesian product of A union C.

iff=if and only iff

Consider that the ordered pair (x,y) is an element of A X (B U C)
which is true iff x is an element of A and y is an element on (B U C)
iff x is an element of A and y is an element of B or y is an element of C
iff x is in A and y is in B or x is in A and y is in C
iff (x,y) is in A X B or (x,y) is in A X C
iff (x,y) is in (A X B) U (A X C)

thus A X (B U C) = (A X B) U (A X C)

Prove that the intersection of A and B is a subset of A

Prove that the intersection of A and B is a subset of A


Suppose there is a x which is in the intersection of A and B. Therefore that x is also in A, and so the intersection of A and B is a subset of A. ~~~~

Let A= x: P(x) and B= x: Q(x) Prove that if for all x: P(x) = > Q(x) then A is a subset of B.

Let A= x: P(x) and B= x: Q(x)Prove that if for all x, P(x) implies Q(x) then A is a subset of B.

Let x be any object and suppose that it is a subset of A. Then P(x) must be true for all x, and since for all x, P(x) implies Q(x) then x is a subset of B. So therefore, A is a subset of B.~~~~

Prove if A is a subset of B, and B is a subset of C, then A is a subset of C.

Prove if A is a subset of B, and B is a subset of C, then A is a subset of C.

Assume A is a subset of B, and B is a subset of C.

Then there is an x in B such that, for every x in B x is in C and there exists an x in A such that, for every x in A x is in B. Thus, for every x in A, x is also in C, and therefore A is a subset of C.

Prove for any set A, A is a subset of A

Prove for any set A, A is a subset of A

Assume we have a set A, and x is an object in that set.

Then x being a subset of A, implies x is a subset of A.

Thus, for all x that is are objects in A, x is an object in A, and so, A is a subset of A. ~~~~

Prove that X = Y where X is a solution to x^2 - 1 = 0 and y = 1,-1

Prove that X = Y where X is a solution to x^2 - 1 = 0 and y = 1,-1

This is a set theory proof. The idea is that if X is a subset of Y, and Y is a subset of X, then X = Y.

So, first we show that Y is a subset of X. And we can do this by simply substituting the values for y in x^2 - 1 = 0. We see that both 1 and -1 work when substituted for X, and so Y is a subset of X.

Now we have to prove that X is a subset of Y. If we factor x^2 - 1 we get (x-1)(x+1) = 0. It is evident that this product is 0 exactly when x=1 or x=-1; so X is a subset of Y.

Now since Y is a subset of X, and X is a subset of Y, then X=Y