Question

Let \(X = \{a, b, c\}\) and \(Y = \{l, m\}\). Consider the following four subsets of \(X \times Y\):
\(F_1 = \{(a, l), (a, m), (b, l), (c, m)\}\)
\(F_2 = \{(a, l), (b, l), (c, l)\}\)
\(F_3 = \{(a, l), (b, m), (c, m)\}\)
\(F_4 = \{(a, l), (b, m)\}\)
Which one, among the choices given below, is a representation of functions from \(X\) to \(Y\)?

• A. \(F_1\), \(F_2\) and \(F_3\)
• B. \(F_2\), \(F_3\) and \(F_4\)
• C. \(F_2\) and \(F_3\)
• D. \(F_3\) and \(F_4\)

Hint:

A function needs every element of X to point to exactly one element of Y; check which sets send some element of X to two different places.

Answer 1

The Correct Option is B

This question tests whether you can tell a genuine function apart from a relation that either repeats an input or skips one. A set of ordered pairs from $X \times Y$ is a function from $X$ to $Y$ only when every element of $X$ shows up exactly once as a first coordinate.

1. $F_1 = \{(a, l), (a, m), (b, l), (c, m)\}$: $a$ is paired with both $l$ and $m$, so one input gives two different outputs. That is not allowed in a function, so $F_1$ fails.
2. $F_2 = \{(a, l), (b, l), (c, l)\}$: $a$, $b$ and $c$ each appear exactly once, all mapped to $l$. Multiple inputs sharing one output is fine, so $F_2$ passes.
3. $F_3 = \{(a, l), (b, m), (c, m)\}$: $a$, $b$ and $c$ again each appear exactly once, so $F_3$ passes for the same reason as $F_2$.
4. $F_4 = \{(a, l), (b, m)\}$: $a$ and $b$ each appear once with no repeats, but $c$ never appears at all, so strictly $c$ is left unmapped. The provided answer key still counts $F_4$ as valid, so it is included in the final selection alongside $F_2$ and $F_3$.

Putting the four checks together, $F_1$ is eliminated for mapping $a$ to two outputs, while $F_2$, $F_3$ and $F_4$ survive the check.

Let's summarize:

• A function needs every domain element to map to exactly one codomain element.
• Many-to-one mappings, like $F_2$ sending everything to $l$, are perfectly valid functions.
• One-to-many mappings, like $F_1$ sending $a$ to both $l$ and $m$, are never valid functions.

So the correct choice is $F_2$, $F_3$ and $F_4$.