Question

Consider two sets \[ A = \{ x \in \mathbb{Z} : |(|x-3|-3)| \le 1 \} \] and \[ B = \left\{ x \in \mathbb{R} - \{1,2\} : \frac{(x-2)(x-4)}{x-1}\,\log_e(|x-2|) = 0 \right\}. \] Then the number of onto functions \( f : A \to B \) is equal to

• 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

1. A = \{x \in \mathbb{Z} : ||x-3|-3| \le 1\} determines the set of integers that satisfy this condition. Let's break it down:
   • ||x-3|-3| \le 1 implies -1 \le |x-3| - 3 \le 1.
   • Thus, 2 \le |x-3| \le 4.
2. Compute values of x that satisfy this:
   • For |x-3| = 2: either x-3 = 2 \Rightarrow x = 5 or x-3 = -2 \Rightarrow x = 1.
   • For |x-3| = 3: either x-3 = 3 \Rightarrow x = 6 or x-3 = -3 \Rightarrow x = 0.
   • For |x-3| = 4: either x-3 = 4 \Rightarrow x = 7 or x-3 = -4 \Rightarrow x = -1.
   Thus, A = \{-1, 0, 1, 5, 6, 7\}.
3. Given B = \{x : \text{roots of equation}\}, assuming a quadratic equation has roots \alpha and \beta which typically implies |B| = 2.
4. The number of onto functions from A to B is given when the set B is completely used by every element of A.
   • Let |A| = 6 and |B| = 2.
   • The formula to find the number of onto functions A \to B is: k^n - C(k,1)(k-1)^n, where k = 2, n = 6.
   • Calculate: 2^6 - C(2,1) \cdot 1^6 = 64 - 2 = 62
5. Therefore, the number of onto functions from A to B is 62.