Question

Sets $A = \{x \in Z : ||x-3|-3| \le 1\$ and $B = \{x : \text{roots of eq}\}$. Number of onto functions $A \to B$.}

• A. 32
• B. 62
• C. 81
• D. 79

Hint:

Number of surjections from size $m$ to size $n$: $\sum_{k=0}^n (-1)^k \binom{n}{k} (n-k)^m$.

Answer 1

The Correct Option is B

For A: $-1 \le |x-3|-3 \le 1 \implies 2 \le |x-3| \le 4$.
$2 \le x-3 \le 4 \implies x \in \{5, 6, 7\}$.
$-4 \le x-3 \le -2 \implies x \in \{-1, 0, 1\}$.
Set A has 6 elements.
For B: $\frac{(x-2)(x-4)}{x-1}\ln|x-2| = 0$. Domain $x \ne 1, 2$.
Roots: $x-4=0 \implies x=4$. $\ln|x-2|=0 \implies |x-2|=1 \implies x=3$ (x=1 rejected).
Set B has 2 elements $\{3, 4\}$.
Total functions = $2^6 = 64$.
Into functions (range size<2) = 2 (all to 3 or all to 4).
Onto functions = $64 - 2 = 62$.