Statements and implications

A statement is a sentence that is either true or false, but not both. “Lagos is in Nigeria” is a true statement and “7 is an even number” is a false one. “Close the door” and “Is it raining?” are not statements: they can’t be true or false. Whether a statement is true or false is its truth value, T or F. Statements are named with small letters such as $p$ and $q$.

The negation of $p$, written ${\sim}p$ (“not $p$”), says the opposite. If $p$ is “Kofi is tall”, then ${\sim}p$ is “Kofi is not tall”. When $p$ is true, ${\sim}p$ is false, and the other way round.

Joining statements

Two statements can be joined into one:

• $p \wedge q$, ”$p$ and $q$” (the conjunction): true only when both are true. “But” means “and” here: with $r$: “it is sunny” and $s$: “it is warm”, “it is sunny but cold” is $r \wedge {\sim}s$.
• $p \vee q$, ”$p$ or $q$” (the disjunction): true when at least one is true, including when both are.
• $p \Rightarrow q$, “if $p$, then $q$” (the implication, also read ”$p$ implies $q$”): false only when $p$ is true and $q$ is false.
• $p \Leftrightarrow q$, ”$p$ if and only if $q$”: true when $p$ and $q$ have the same truth value.

Try it

Fill in “if $p$ then $q$” row by row. Think of $p \Rightarrow q$ as a promise: “if Ada scores 80, she gets a prize”. The promise is broken only if she scores 80 and gets no prize. When she doesn’t score 80, the promise isn’t tested, so it counts as true.

Converse, inverse and contrapositive

From an implication $p \Rightarrow q$ you can make three others:

• the converse $q \Rightarrow p$: swap the two parts;
• the inverse ${\sim}p \Rightarrow {\sim}q$: negate both parts;
• the contrapositive ${\sim}q \Rightarrow {\sim}p$: swap and negate.

Only the contrapositive means the same as $p \Rightarrow q$. “If Ada scores 80, she gets a prize” and “if Ada gets no prize, she didn’t score 80” say the same thing. The converse, “if Ada gets a prize, she scored 80”, doesn’t follow: she might have won it another way. So from $p \Rightarrow q$:

Pick “The four implications” on the board above to see it in the truth table: the contrapositive’s column is the same as $p \Rightarrow q$‘s, row for row.

Your turn