A-Level Mathematical argument and notation Practice Questions

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’).

Read the argument

‘Hence’ signals that the second line follows logically from the first.

Check the deduction

Dividing both sides by $3$ genuinely gives $x=4$, so the conclusion does follow.

Conclude

‘Hence’ is correct because $x=4$ is a valid consequence of $3x=12$.

Forward direction

If a product of two real numbers is zero, at least one factor must be zero (the zero-product property).

Reverse direction

If either factor is zero then the product is zero, so the reverse holds too.

Both directions hold

Since each side implies the other, the link is a full equivalence.

Why it matters

This equivalence is exactly what lets us solve $(x-2)(x-3)=0$ by setting each bracket to zero.

Note the real-number condition

This relies on the reals having no ‘zero divisors’; it would fail in some other number systems.

Conclude

The correct connective is $\Leftrightarrow$.

Read the claim

It claims that a bigger number always has a bigger square.

Try positive numbers

For positive values it seems to work.

Test with negatives

The danger with squaring is negative numbers, so test $x=1$, $y=-2$.

Check the hypothesis

Here $x>y$ is true.

Check the conclusion

But $x^2=1$ and $y^2=4$.

Compare the squares

The conclusion $x^2>y^2$ is FALSE here.

Identify the counterexample

This single case breaks the implication.

Explain the reason

Squaring depends on magnitude, not order, so ordering is not preserved when negatives are involved.

State the corrected version

The correct statement compares magnitudes, not the numbers themselves.

Conclude

The implication is false; a counterexample such as $x=1,y=-2$ disproves it.

Symbolise the argument

This is a valid deductive form (modus ponens).

Check the logical form

If the premises were true, the conclusion would follow — so the FORM is valid.

Test the first premise

Examine whether ‘all primes are odd’ is actually true.

Find a counterexample

$2$ is a prime number that is even, so the first premise is FALSE.

Distinguish validity from truth

Validity is about the structure; it does not guarantee the premises are true.

Define soundness

An argument is sound only if it is valid AND all premises are true.

Classify this argument

Here the form is valid, but a false premise makes it unsound, so the conclusion can be false.

Explain the false conclusion

A valid argument from a false premise can produce a false conclusion.

State precisely

The argument is valid in form but unsound because the premise is false.

Define a valid argument

Validity means: IF the premises were true, the conclusion would have to be true.

Define a sound argument

Soundness additionally requires the premises actually to be true.

Identify the modus ponens form

The argument follows the valid modus ponens pattern.

Test the questionable premise

We must check whether ‘all primes are odd’ is true.

Produce the counterexample

$2$ is prime yet even, so the premise is false.

Draw the distinction

Hence the argument is valid in form but unsound, so a false conclusion is possible.