CMI 2023 | Data Science | Part A | Question: 16

Question

Let $X, Y, Z$ be finite sets. Which of the following statement(s) is/are true?

• A. Let $f: X \rightarrow Y, g: Y \rightarrow Z$ be functions. such that the composite function $g \circ f: X \rightarrow Z$ is onto. Then both $g, f$ have to be onto.
• B. Assume the number of elements of $X$, denoted as $|X|$, is strictly less than $|Y|$ and also assume that $|X| \leq|Z|$. There exists a one-one function $f: X \rightarrow Y$ and a $g: Y \rightarrow Z$ which is not one-one such that $g \circ f: X \rightarrow Z$ is one-one.
• C. Let $i d_{X}$ be the identity function on $X$, i.e. $i d_{X}(x)=x$ for all $x \in X$. Let $f: X \rightarrow Y$ be a function. Then there exists a function $g: Y \rightarrow X$ such that $g \circ f=i d_{X}$ exists.
• D. Denote by $i d_{Y}$ the identity function on $Y, i d_{Y}(y)=y$ for all $y \in Y$. Then there exists $g: Y \rightarrow X$ such that $f \circ g=i d_{Y}$ if and only if $f$ is onto.