8th grade Q2

To illustrate the truthfulness of an if-then statement ($p\to q$), you can use a truth table to evaluate all possible combinations of the hypothesis ($p$) and conclusion ($q$).

The truth table shows when the statement is true or false based on the truthfulness of the hypothesis and conclusion.

Truth Table for an If-Then Statement

The statement "If $p$, then $q$" ($p\to q$) is:

• False only when $p$ is true and $q$ is false.
• True in all other cases.

This can be represented using a truth table.

Explanation

1. Row 1: If $p$ is true and $q$ is true, the statement $p\to q$ is true.
2. Row 2: If $p$ is true and $q$ is false, the statement $p\to q$ is false (the only case where $p\to q$ is false).
3. Row 3: If $p$ is false, $p\to q$ is true regardless of $q$.
4. Row 4: If $p$ is false and $q$ is false, $p\to q$ is true.

The third row may surprise you. Think of it like this: There are situations where the outcome still happens, even though the assumption didn't.

For instance:

"If a shape has 4 sides, then it is a polygon".

Now consider a pentagon:

• The shape has 5 sides. The assumption is false.
• The shape is a polygon. The conclusion is true.

Example

Statement: "If I study, then I will pass the test."

• Hypothesis: "I study" ($p$).
• Conclusion: "I will pass the test" ($q$).

Note

• The statement is false only if you study ($p=T$) and still fail the test ($q=F$).
• In all other scenarios, the statement is true, regardless of whether you studied or not.

Key Points:

• Truth tables illustrate the conditions under which an if-then statement is true or false.
• Understanding the truth table provides a systematic way to analyze the truthfulness of a statement.