Let X and Y be sets, let A and B be any subsets of X, and let F be a function from X to

Chapter 7, Problem 40

(choose chapter or problem)

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 \(F(A) \cup F(B) \subseteq F(A \cup B)\).

Proof: Let y be any element in \(F(A) \cup F(B)\). [We must show that y is in \(F(A \cup B)\).] By definition of union, (a).

Case 1, y ∈ F(A): In this case, by definition of F(A), y = F(x) for (b) \(x \in A\). Since \(A \subseteq A \cup B\), it follows from the definition of union that \(x \in\) (c). Hence, y = F(x) for some \(x \in A \cup B\), and thus, by definition of \(F(A \cup B)\), \(y \in\) (d).

Case 2, \(y \in\) F(B): In this case, by definition of F(B), (e) \(x \in B\). Since \(B \subseteq A \cup B\) it follows from the definition of union that ( f ).

Therefore, regardless of whether \(y \in F(A)\) or \(y \in F(B)\), we have that \(y \in F(A \cup B)\) [as was to be shown].

Text Transcription:

F(A) cup F(B) subseteq F(A cup B)

F(A) cup F(B)

F(A cup B)

x in A

A subseteq A cup B

x in

x in A cup B

F(A cup B)

y in

y in

x in B

B subseteq A cup B

y in F(A)

y in F(B)

y in F(A cup B)