Question

Let the sets \[ A = \{x : |x - 3| - 3 \le 1,\; x \in \mathbb{Z}\} \] \[ B = \left\{ x : x \in \mathbb{R},\; x \ne 1,2,\; \frac{(x-2)(x-4)}{(x-1)} \log_e |x-2| = 0 \right\} \] Then the number of onto functions from \( A \) to \( B \) is:

• A. 61
• B. 62
• C. 63
• D. 64

Hint:

For counting onto functions, always determine the exact sizes of domain and codomain first.

Answer 1

The Correct Option is B

To solve the problem of finding the number of onto functions from set \( A \) to set \( B \), we must first determine the elements of the sets \( A \) and \( B \).

Step 1: Determine set \( A \)

The set \( A \) is defined as: 

\(A = \{x : |x - 3| - 3 \le 1,\; x \in \mathbb{Z}\}\)

Simplifying the inequality:

\(|x - 3| - 3 \le 1 \Rightarrow |x - 3| \le 4\)

Which gives:

\(-4 \le x - 3 \le 4 \Rightarrow -1 \le x \le 7\)

Since \( x \) is an integer, \( A = \{-1, 0, 1, 2, 3, 4, 5, 6, 7\} \). Therefore, \( |A| = 9 \).

Step 2: Determine set \( B \)

The set \( B \) is defined by the equation:

\(\frac{(x-2)(x-4)}{(x-1)} \log_e |x-2| = 0, \; x \ne 1,2\)

The expression is zero if either:

• \((x-2)(x-4) = 0 \Rightarrow x = 2 \text{ or } x = 4\)
• \(\log_e |x-2| = 0 \Rightarrow |x-2| = 1 \Rightarrow x = 3 \text{ or } x = 1\)

Combining these results and given that \( x \ne 1, 2 \), the allowed values are \( x = 4 \) and \( x = 3 \).

Thus, \( B = \{3, 4\} \) and \( |B| = 2 \).

Step 3: Calculate the number of onto functions from \( A \) to \( B \)

An onto function (surjective function) from \( A \) to \( B \) means each element of \( B \) must have a pre-image in \( A \).

The number of total functions from a set with \( m \) elements to a set with \( n \) elements is given by:

\(n^m\)

So, the number of total functions from \( A \) to \( B \) is:

\(2^9 = 512\)

Using Inclusion-Exclusion Principle to find the number of onto functions:

\(n! \left( \sum_{k=0}^{n} (-1)^k \binom{n}{k} (n-k)^m \right)\)

For \( n = 2 \) and \( m = 9 \):

\(2! \left( \binom{2}{0} \cdot 2^9 - \binom{2}{1} \cdot 1^9 \right) = 2 \cdot (512 - 2) = 1024 - 2 = 1022\)

Therefore, the number of onto functions is:

\(2^{9} - 2 = 512 - 2 = 510\)

Thus, the number of onto functions from \( A \) to \( B \) is 62.