Question 8
Consider two sets $$A=\left\{x\in Z:|(|x-3|-3)\leq1\right\}$$ and $$B=\left\{x \in \mathbb R-\left\{1,2\right\}:\frac{(x-2)(x-4)}{x-1}\log_{e}(|x-2|)=0 \right\}$$. Then the number of onto functions $$f:A\rightarrow B$$ is equal to
Solution
A = {x ∈ Z : ||x-3|-3| ≤ 1}. |x-3| ∈ [2,4], so x-3 ∈ [-4,-2]∪[2,4], x ∈ {-1,0,1,5,6,7}. |A| = 6.
B = {x ∈ R\{1,2}: (x-2)(x-4)ln|x-2|/(x-1) = 0}. Solutions: x=4 (from x-4=0) or |x-2|=1 giving x=3 (x=1 excluded). So B = {3,4}. |B| = 2.
Onto functions from 6-element set to 2-element set: 2⁶ - 2 = 62.
The answer is Option 3: 62.
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