Let f: A → B be a function, and let E and F be subsets of A. Consider the…
2001
Let f: A → B be a function, and let E and F be subsets of A. Consider the following statements about images.
S1: f(E ∪ F) = f(E) ∪ f(F)
S2: f(E ∩ F) = f(E) ∩ f(F)
Which of the following is true about S1 and S2?
- A.
Only S1 is correct
- B.
Only S2 is correct
- C.
Both S1 and S2 are correct
- D.
Neither S1 nor S2 is correct
Attempted by 26 students.
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Correct answer: A
The correct answer is Option A (Only S1 is correct). For any y in f(E ∪ F), there is some x in E ∪ F with f(x) = y. Then x is in E or in F, so y is in f(E) ∪ f(F). Hence S1 is always true. For S2, only f(E ∩ F) ⊆ f(E) ∩ f(F) is guaranteed. Equality can fail: let E = {1}, F = {2}, and f(1) = f(2) = b. Then E ∩ F is empty, so f(E ∩ F) is empty, but f(E) ∩ f(F) = {b}. Therefore S2 is not always correct.