Assume that sets A and B are subsets of a universal set U Pr

Assume that sets A and B are subsets of a universal set U. Prove that if A C B, then B^c C A^C.

Since A is a subset of B, hence we can write

A U B = B (since A is subset of B, hence all the elements of A are already counted in B)

A (int) B = A (since A is a subset of B, hence all the elements of A are also present in B)

Bc = U - B, where U represents the universal set

Similarly Ac = U - A

we need to prove that Bc is a subset of Ac

Let us assume that an element p belongs to Bc

This implies the element belongs to the set (U-B)

which implies that the elements p belongs to U and p doesn\'t belong to B

since A is subset of B, hence the element doesn\'t belong to A

therefore we can write that element p belongs to U and p doesn\'t belong to A from the first relation

Hence, p belongs to (U-A)

Without loss of generality, we can say that given thing holds for every p, hence the set Bc is a subset of Ac