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.
None of S₁ and S₂ are correct
Attempted by 32 students.
Show answer & explanation
Correct answer: A
For S1, if y is in f(E union F), then y = f(x) for some x in E union F. Hence x is in E or x is in F, so y is in f(E) union f(F). The reverse inclusion is also immediate, so S1 is true. For S2, we always have f(E intersection F) subset of f(E) intersection f(F), but equality need not hold if different elements have the same image. For example, if E = {a}, F = {b}, a is not equal to b, and f(a) = f(b), then E intersection F is empty but f(E) intersection f(F) is not empty. Thus S2 is false in general. Hence only S1 is correct.
A video solution is available for this question — log in and enroll to watch it.