Let X and Y be sets, let A and B be any subsets of X, and let F be a function from X to Y. Fill in the blanks in the following proof that F ( A ) ∪ F ( B ) ⊆ F ( A ∪ B ) .

To Fill:

The following proof that F(A)∪F(B)⊆F(A∪B).

Given information:

Let X and Y be sets, let A and B be any subsets of X and let F be a function from X to Y, and the incomplete proof of the statement F(A)∪F(B)⊆F(A∪B).

Calculation:

Given:

X and Y are sets

A and B are subsets of X F is a function from X to Y

To proof: F(A)∪F(B)⊆F(A∪B)

PROOF:

Let y∈F(A)∪F(B). By the definition of the union, we then know that

1. y∈F(A) or y∈F(B)

First case: y∈F(A)

By the definition of the image, there exists (b) some x∈A such that y=F(x).

Since x∈A and since A⊆A∪B :

x∈(c)A∪B

Thus we then obtain y=F(x) for some x∈A∪B and thus y lies in the image of A∪B