Which of the following function f: Z X Z → Z is not onto?a.f(a, B) = a – bb.f(a, B) = |b|c.f(a, B) = a + bd.f(a, B) = a

To determine which of the given functions $f: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}$ is not onto, we need to analyze each function to see if every integer in $\mathbb{Z}$ can be obtained as an output.

1. Analyze Each Function

a. $f(a, b) = a - b$

For any integer $k$, we can choose $a = k + b$ where $b$ is any integer. Therefore, for $k \in \mathbb{Z}$, we can find $a$ and $b$ such that $f(a, b) = k$. This function is onto.

b. $f(a, b) = |b|$

The output is always a non-negative integer because the absolute value of $b$ is taken. Therefore, the function can never output negative integers. Hence, this function is not onto.

c. $f(a, b) = a + b$

For any integer $k$, we can simply choose $a = k$ and $b = 0$ (or $b = k - a$, as needed). This means we can reach any integer $k$, making this function onto.

d. $f(a, b) = a$

This function outputs $a$, which can take any integer value from $\mathbb{Z}$ depending on the input. Hence, it is onto.

2. Final Analysis

From the analysis above:

• Function (a): Onto
• Function (b): Not onto (only outputs non-negative integers)
• Function (c): Onto
• Function (d): Onto

Final Answer

The function that is not onto is b. $f(a, b) = |b|$ because it cannot produce negative integers in its range.