Let f:X→Y be a function, A and B be the subsets of X. Which of the following…
Let f:X→Y be a function, A and B be the subsets of X. Which of the following conditions are true?I.If A⊆B then f(A)⊆f(B)II.fA∩fB⊆f(A∩B)
Answer: A. Only I is true — Answer: Only the first statement is true. Proof that the first statement is true: If A ⊆ B and y ∈ f(A), then there exists x ∈ A with f(x) = y. Since A ⊆ B,…
- A.
Only I is true
- B.
I and II are true
- C.
II is true, I is false
- D.
Both are false
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Correct answer: A
Answer: Only the first statement is true.
Proof that the first statement is true: If A ⊆ B and y ∈ f(A), then there exists x ∈ A with f(x) = y. Since A ⊆ B, that x is also in B, so y ∈ f(B). Therefore f(A) ⊆ f(B).
The second statement is false (counterexample):
Let X = {1, 2}, Y = {a}, and define f by f(1) = a, f(2) = a.
Take A = {1} and B = {2}. Then f(A) = {a} and f(B) = {a}, so f(A) ∩ f(B) = {a}.
But A ∩ B = ∅, so f(A ∩ B) = ∅. Hence f(A) ∩ f(B) is not contained in f(A ∩ B).
Related correct inclusion: In general we always have f(A ∩ B) ⊆ f(A) ∩ f(B), but the reverse inclusion need not hold.