TMUA Logic: Implication, Converse and Contrapositive

Implication and its relatives

• Statement: if $P$ then $Q$ ($P\Rightarrow Q$).
• Converse: if $Q$ then $P$. Not equivalent to the statement.
• Contrapositive: if not $Q$ then not $P$. Always equivalent to the statement.

Wording

'$P$ only if $Q$' means $P\Rightarrow Q$. '$P$ if $Q$' means $Q\Rightarrow P$. '$P$ if and only if $Q$' means both.

Necessary and sufficient

If $P\Rightarrow Q$ then $P$ is sufficient for $Q$ and $Q$ is necessary for $P$.

Worked examples

Common mistakes

• Treating the converse as if it were equivalent to the original statement.
• Reading '$P$ only if $Q$' as '$Q\Rightarrow P$'.
• Mixing up necessary and sufficient.
• Treating 'not $P$ implies not $Q$' (the inverse) as the contrapositive.

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