G.RL.1.2: Reasoning and If-Then Statements

How Do We Figure Things Out?

In math (and life!), we use different ways to reach conclusions. We look for patterns (inductive reasoning) or use established facts and rules (deductive reasoning). We also use "if-then" statements to explore logical connections.

This lesson dives into these ways of thinking and the structure of logical arguments.

1. Inductive vs. Deductive Reasoning

Two main ways we reason:

Inductive Reasoning

Making conclusions based on observing patterns or specific examples.

Example:

Every swan I've ever seen is white.
Conclusion (inductive): All swans are white.

Warning: Inductive conclusions might be wrong! (There are black swans).

Deductive Reasoning

Making conclusions based on established facts, definitions, rules, postulates, or theorems.

Example:

Fact 1: All squares are rectangles.
Fact 2: Figure ABCD is a square.
Conclusion (deductive): Figure ABCD is a rectangle.

If the facts/rules are true, the deductive conclusion MUST be true.

Practice: Inductive or Deductive?

Read each statement and decide if the conclusion uses inductive (pattern-based) or deductive (fact/rule-based) reasoning. Click the button!

1. The last three times I went to the park, it was sunny. Therefore, it will be sunny the next time I go to the park.

2. All students in this class passed the test. Maria is in this class. Therefore, Maria passed the test.

3. Parallel lines never intersect. Line $m$ is parallel to line $n$. Therefore, line $m$ and line $n$ never intersect.

4. I flipped a coin four times and got heads each time. Therefore, the next flip will also be heads.

2. Conditional Statements (If-Then)

These are statements written in the form "If P, then Q", often symbolized as $P \rightarrow Q$.

Example: If an animal is a poodle, then it is a dog.

Hypothesis (P): The part following "If". (An animal is a poodle)
Conclusion (Q): The part following "Then". (It is a dog)

3. Related Conditional Statements

From an original conditional statement ($P \rightarrow Q$), we can form three related statements by switching or negating the hypothesis and conclusion.

Converse (Switch P and Q)

Form: If Q, then P. ($Q \rightarrow P$)

Example: If an animal is a dog, then it is a poodle. (Is this true? Not necessarily!)

Inverse (Negate P and Q)

Form: If not P, then not Q. ($\sim P \rightarrow \sim Q$) (The ~ symbol means "not")

Example: If an animal is not a poodle, then it is not a dog. (Also not necessarily true!)

Contrapositive (Switch AND Negate P and Q)

Form: If not Q, then not P. ($\sim Q \rightarrow \sim P$)

Example: If an animal is not a dog, then it is not a poodle. (This IS true!)

Key Idea: A conditional statement and its contrapositive are logically equivalent – they are either both true or both false. The converse and inverse are also logically equivalent to each other (but not necessarily to the original statement).

Match the Logic!

Given the statement: "If it is raining ($P$), then the ground is wet ($Q$)."

Click a statement type on the left, then click its matching example on the right.

Conditional ($P \rightarrow Q$)

Converse ($Q \rightarrow P$)

Inverse ($\sim P \rightarrow \sim Q$)

Contrapositive ($\sim Q \rightarrow \sim P$)

If it is not raining, then the ground is not wet.

If it is raining, then the ground is wet.

If the ground is not wet, then it is not raining.

If the ground is wet, then it is raining.

Ways of Thinking & If-Then Logic