Q.Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.

Concept: Relation Properties — A function requires each input to have exactly one output. Injectivity means distinct inputs map to distinct outputs; surjectivity means every possible output is used.

(i) Each person has exactly one mother, so this is a function.

• Injective? No — two siblings share the same mother, so distinct $x$ can map to the same $y$.
• Surjective? No — not every woman is the mother of some person in the domain (e.g., childless women). …

Let’s first get clear on what a function is. A relation from set $A$ to set $B$ is a function if every element of $A$ is related to exactly one element of $B$. That’s the core rule: one input, one output. Once we decide if it’s a function, we then check injectivity (one-to-one: different inputs map to different outputs) and surjectivity (onto: every element of $B$ is used as an output).

(i) $\{(x, y) : x \text{ is a person}, \ y \text{ is the mother of } x\}$

1. Is it a function?

Every person has exactly one biological mother. So for each person $x$, there is precisely one $y$ (the mother). That satisfies the definition: each input $x$ gives a unique output $y$.

Yes, this is a function. Let’s call it $f: \text{People} \to \text{Women}$.

2. Is it injective?

Injectivity means: if $f(x_1) = f(x_2)$, then $x_1 = x_2$. But here, two different people can have the same mother (siblings). For example, if $x_1$ and $x_2$ are siblings, $f(x_1) = f(x_2)$ but $x_1 \neq x_2$.

So the function is not injective.

3. Is it surjective?

Surjectivity means: every woman in the codomain must be the mother of at least one person. But not every woman is a mother — some women have no children. So the range (set of all mothers) is a proper subset of all women.

Thus the function is not surjective.

(ii) $\{(a, b) : a \text{ is a person}, \ b \text{ is an ancestor of } a\}$

1. Is it a function? …

Method: Deciding function first, then injective/surjective

For "is this set of pairs a function, and if so is it injective/surjective," always test the function condition before the other two.

Steps

Step 1: Function test — unique image per input

A relation is a function only if each input has exactly one output. A "one-to-many" rule (like "ancestor of") fails immediately and the injective/surjective questions do not even apply.

Step 2: Injective (one-one) test

Ask whether two different inputs can share the same output. If yes (e.g. siblings share a mother), it is not injective.

Step 3: Surjective (onto) test …

Common Mistakes

Mistake 1: Testing injective/surjective on something that is not a function

Why it's wrong: "ancestor of" assigns many outputs to one person, so it is not a function and those properties are undefined for it. Correct approach: confirm single-valuedness first; if it fails, conclude "not a function" and stop.

Mistake 2: Believing "mother of" is injective

Why it's wrong: two siblings have the same mother, so distinct inputs share an output. Correct approach: a shared image among different inputs breaks injectivity. …