Exercise · ZW-IMAGE-AND-PREIMAGE-007 · ★★☆ Intermediate
Let \(f\colon A\to B\) and \(X_1,X_2\subseteq A\). Which statement about images is correct?
Choose one option.
Show answer
- A. \(f(X_1\cup X_2)=f(X_1)\cup f(X_2)\)
Explanation
The image of a union is the union of the images, since each value comes from a point in \(X_1\) or in \(X_2\). For intersections and differences equality fails: for \(f(x)=x^2\), \(X_1=\{-1\}\) and \(X_2=\{1\}\), the intersection of the sets is empty but \(f(X_1)\cap f(X_2)=\{1\}\).
Review the concept: Image and preimage
Given \(f\colon A\to B\), the image of \(X\subseteq A\) is the set of values \(f(x)\) with \(x\in X\), and the preimage of \(Y\subseteq B\) is the set of \(a\in A\) whose value lies in \(Y\).
\[f(X)=\{f(x)\colon x\in X\},\qquad f^{-1}(Y)=\{a\in A\colon f(a)\in Y\}\]
More practice problems: image and preimage
- ZW-IMAGE-AND-PREIMAGE-001 What is the image of X⊆ A under fcolon A→ B?
- ZW-IMAGE-AND-PREIMAGE-002 Which subset of A is the preimage of Y⊆ B?
- ZW-IMAGE-AND-PREIMAGE-003 If fcolon A→ B is surjective, what is f(A)?
- ZW-IMAGE-AND-PREIMAGE-004 For Y⊆ B, what is always true of f(f^-1(Y))?
- ZW-IMAGE-AND-PREIMAGE-005 Let f(x)=x^2 from ℝ to ℝ. What is f([-1,2])?
- ZW-IMAGE-AND-PREIMAGE-006 With f(x)=x^2 from ℝ to ℝ, what is f^-1([1,4])?
- ZW-IMAGE-AND-PREIMAGE-008 What is f^-1(\y\) when y is not in f(A)?
- ZW-IMAGE-AND-PREIMAGE-009 Let fcolonℝ→ℝ, f(x)=x^2-2x. What is f^-1(\-1\)?