Mathematical argument and notation Worked Solutions — A-Level Maths

Read the statement

We are told the link only works one way: knowing $x=3$ lets us conclude $x^2=9$, but not the other way round.

Test the forward direction

If $x=3$ then squaring gives $9$, so the arrow from left to right is true.

Test the reverse direction

The reverse fails because $x$ could be $-3$. So this is NOT a two-way $\Leftrightarrow$.

Choose the symbol

A one-way link uses the implication arrow $\Rightarrow$ (‘implies’).

State the converse

The converse swaps the two sides of the arrow: we now assume $x^2=9$ and ask whether $x=3$ must follow.

Look for a counterexample

A counterexample is one case that breaks the claim. Try $x=-3$: squaring gives $9$.

Check the conclusion fails

Here $x^2=9$ is true but $x=3$ is false, so the converse does not always hold.

Conclude

Because a single counterexample exists, the converse statement is false.

Test forward

If $n$ is even then adding $1$ makes it odd, so the forward arrow holds.

Test reverse

If $n+1$ is odd then $n$ must be even, so the reverse arrow also holds.

Both directions hold

When both directions are true we use the equivalence symbol $\Leftrightarrow$ (‘if and only if’).

Conclude

This is a genuine two-way equivalence.

Recall the symbol

The three dots in a triangle pointing up is standard shorthand used in written arguments.

Give its meaning

It introduces a conclusion that follows from the lines above it.

Contrast

Do not confuse it with $\because$ (dots pointing down), which means ‘because’.

Test sufficiency

Every multiple of $4$ is even, so the condition guarantees evenness: it is sufficient.

Test necessity

A number can be even without being a multiple of $4$ (e.g. $6$), so it is not necessary.

Conclude

It forces evenness but is not required for it.