Quick questions on Mathematical arguments and proof explained: H2 Further Mathematics

A statement of the form "if $P$ then $Q$" is the implication $P \Rightarrow Q$; $P$ is the hypothesis and $Q$ the conclusion. Two related statements matter:

A direct proof of $P \Rightarrow Q$ assumes $P$ and reasons forward through valid steps to reach $Q$. Most algebraic and identity proofs are direct: start from the definitions or given facts and manipulate to the conclusion.

Since $\lnot Q \Rightarrow \lnot P$ is equivalent to $P \Rightarrow Q$, you may prove the contrapositive instead. This is useful when assuming $\lnot Q$ gives more to work with than assuming $P$, as in "if $n^2$ is even then $n$ is even", where assuming $n$ odd is concrete.

To prove a statement $S$ by contradiction, assume $S$ is false and derive a logical impossibility (a contradiction with a known fact or with the assumption itself). The contradiction shows the assumption was untenable, so $S$ must be true. The classic example is the irrationality of $\sqrt{2}$.

Verifying a statement for $n = 1, 2, 3$ is not a proof; it could fail at the next value. Give a general argument.

A valid counterexample must meet every condition of the statement and yet fail the conclusion.

State the contrapositive of "if it is raining then the ground is wet". [1 mark]

Explain why one counterexample is enough to disprove "every prime is odd". [2 marks]

Outline how you would begin a proof by contradiction that there is no largest integer. [2 marks]