Q.Let $X = \{1, 2, 3\}$ and $Y = \{4, 5\}$. Find whether the following subsets of $X \times Y$ are functions from $X$ to $Y$ or not.

Testing whether a relation is a function. A subset of $X \times Y$ is a function from $X$ to $Y$ only if every element of $X$ appears exactly once as a first coordinate (no element missing, and none paired with two different outputs). We scan the first coordinates of each set.

1. $f = \{(1,4),(1,5),(2,4),(3,5)\}$: the input $1$ appears twice, mapped to both $4$ and $5$ — two outputs for one input. Not a function.
2. $g = \{(1,4),(2,4),(3,4)\}$: each of $1,2,3$ appears exactly once (all map to $4$, which is allowed). Is a function. …

The Core Idea: What Makes a Relation a Function?

A function from set $X$ to set $Y$ is a special kind of relation — a subset of $X \times Y$ — with two ironclad rules:

1. Every element of $X$ must appear as a first coordinate (no element left behind).
2. Each element of $X$ must pair with exactly one element of $Y$ (no splitting).

Think of it like a vending machine: for every button you press (an $x$), you must get one and only one snack (a $y$). If a button gives two different snacks, or if a button gives nothing, it’s broken — not a function.

Here, $X = \{1, 2, 3\}$ has three elements, and $Y = \{4, 5\}$ has two. So a valid function will have exactly three ordered pairs — one for each $x$ — and the second coordinate can be $4$ or $5$.

Checking Each Subset

1. $f = \{(1, 4), (1, 5), (2, 4), (3, 5)\}$

Look at $x = 1$: it appears twice — once with $4$ and once with $5$. That violates rule 2: one input cannot map to two different outputs.

Also, $f$ has four pairs, but $X$ has only three elements — a red flag that some $x$ is repeated.

Conclusion: $f$ is not a function.

2. $g = \{(1, 4), (2, 4), (3, 4)\}$

Every element of $X$ appears exactly once: $1 \to 4$, $2 \to 4$, $3 \to 4$. …

Method: Deciding whether a subset of $X\times Y$ is a function

A subset of $X\times Y$ is a function from $X$ to $Y$ only if it satisfies two conditions together.

Steps

Step 1: Existence — every $x\in X$ must appear

Check that each element of $X$ occurs as a first coordinate at least once. If any $x$ is missing, it has no image and the set is not a function.

Step 2: Uniqueness — no $x$ repeats with different images

Check that no element of $X$ appears as a first coordinate more than once with two different second coordinates. A repeated input with two outputs disqualifies it. …

Common Mistakes

Mistake 1: Allowing one input to have two outputs

Why it's wrong: $f$ with both $(1,4)$ and $(1,5)$ assigns $1$ two images, violating uniqueness. Correct approach: reject any set where an input repeats with different outputs.

Mistake 2: Overlooking a missing input

Why it's wrong: $k=\{(1,4),(2,5)\}$ leaves $3\in X$ unmapped, so it is not a function on $X$. Correct approach: confirm every element of $X$ appears as a first coordinate. …