Investigation 1: Logic and Truth Tables

New Concept

A logic statement can be written in symbolic form as $p \to q$. "$p$ implies $q$." This is an example of logical implication.

Why it matters

Logic provides the rigorous framework that underpins all mathematical proofs, turning algebraic steps into a verifiable sequence. Mastering this structure allows you to construct airtight arguments and deconstruct complex problems, a crucial skill in advanced mathematics and computer science.

What’s next

Next, you’ll use truth tables to explore the conditions under which different logical statements are true, false, or even logically equivalent.

Property

An 'if-then' statement, written as $p \to q$, means '$p$ implies $q$'. The entire statement is considered true in all cases except one: it is only false when the first part, $p$, is true and the second part, $q$, is false.

Example 1: Statement 'If a creature is a spider ($p$), then it has eight legs ($q$)' is $p \to q$. This is only false if you find a spider that does not have eight legs.
Example 2: Statement 'If you get an A on the test ($p$), we will celebrate ($q$)' is $p \to q$. The promise is only broken if you get an A and we do not celebrate.

Think of this as a one-way promise. The statement 'if p, then q' only breaks if the first part (p) happens but the promised second part (q) doesn't. For example, 'If you clean your room, I'll give you ten dollars.' The only way I've lied is if you actually clean your room and I don't pay up.

Property

The converse of a logical implication $p \to q$ is the statement $q \to p$. It is formed by swapping the hypothesis and the conclusion.

Example 1: Statement: 'If it is snowing ($p$), then the temperature is below freezing ($q$).' $p \to q$. Converse: 'If the temperature is below freezing ($q$), then it is snowing ($p$).' $q \to p$. (False, it can be cold but not snowing).
Example 2: Statement: 'If an animal is a whale ($p$), then it lives in the ocean ($q$).' Converse: 'If an animal lives in the ocean ($q$), then it is a whale ($p$).' (False, fish and dolphins also live in the ocean).

This is like reversing a statement to see if it's still true, but be careful—it often isn't! 'If it's a cat, then it's a mammal' is true. But the converse, 'If it's a mammal, then it's a cat,' is totally false (hello, dogs and whales!). Always test the converse; don't assume it works just because the original did.