Conditional Statements Notes Worksheet

Let's solve the problem step by step.

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Step 1: Fill in the Table

We are given a table about conditional statements and need to complete the missing parts.

#### 1. Conditional Statement
- Definition: A statement with a hypothesis and a conclusion. The statement contains the phrase "if, then".
- Symbol Form: $ p \rightarrow q $
- Example: If I eat a whole pizza, then I will be full.

✔ Already completed correctly.

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#### 2. Converse
- Definition: This statement switches the hypothesis and conclusion.
- Symbol Form: $ q \rightarrow p $
- Example: If I will be full, then I ate a whole pizza.

> *Explanation*: The converse swaps the "if" and "then" parts.

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#### 3. Inverse
- Definition: This statement negates the hypothesis and conclusion.
- Symbol Form: $ \sim p \rightarrow \sim q $
- Example: If I did not eat a whole pizza, then I will not be full.

> *Explanation*: Both the hypothesis and conclusion are negated.

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#### 4. Contrapositive
- Definition: This statement negates and switches the hypothesis and conclusion.
- Symbol Form: $ \sim q \rightarrow \sim p $
- Example: If I am not full, then I did not eat a whole pizza.

> *Explanation*: Negate both and reverse their order.

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✔ Completed Table:

| TYPE OF STATEMENT | DEFINITION | SYMBOL FORM | EXAMPLE |
|-------------------|----------|-------------|---------|
| CONDITIONAL | A statement with a hypothesis and a conclusion. The statement contains the phrase "if, then". | $ p \rightarrow q $ | If I eat a whole pizza, then I will be full. |
| CONVERSE | This statement switches the hypothesis and conclusion. | $ q \rightarrow p $ | If I will be full, then I ate a whole pizza. |
| INVERSE | This statement negates the hypothesis and conclusion. | $ \sim p \rightarrow \sim q $ | If I did not eat a whole pizza, then I will not be full. |
| CONTRAPOSITIVE | This statement negates and switches the hypothesis and conclusion. | $ \sim q \rightarrow \sim p $ | If I am not full, then I did not eat a whole pizza. |

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Step 2: Example Problem

Given conditional statement:
"If a number is even, then it is evenly divisible by 2."

Let’s identify:
- $ p $: A number is even
- $ q $: It is evenly divisible by 2

So: $ p \rightarrow q $

Now find:

#### Converse: $ q \rightarrow p $
"If a number is evenly divisible by 2, then it is even."

✔ This is true — because any number divisible by 2 is, by definition, even.

#### Inverse: $ \sim p \rightarrow \sim q $
"If a number is not even, then it is not evenly divisible by 2."

✔ This is true — odd numbers are not divisible by 2.

#### Contrapositive: $ \sim q \rightarrow \sim p $
"If a number is not evenly divisible by 2, then it is not even."

✔ This is true — if a number isn't divisible by 2, it must be odd (i.e., not even).

> 💡 Note: The original statement, its contrapositive, and the inverse are all logically equivalent in this case (all true). However, the converse is also true here, but that’s not always the case.

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✔ Final Answers for the Example:

Converse:
*If a number is evenly divisible by 2, then it is even.*
→ True

Inverse:
*If a number is not even, then it is not evenly divisible by 2.*
→ True

Contrapositive:
*If a number is not evenly divisible by 2, then it is not even.*
→ True

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📌 Summary of Key Concepts:

- Conditional: $ p \rightarrow q $
- Converse: $ q \rightarrow p $ → Not necessarily logically equivalent
- Inverse: $ \sim p \rightarrow \sim q $ → Not necessarily logically equivalent
- Contrapositive: $ \sim q \rightarrow \sim p $ → Logically equivalent to the original

✔ In logic, a conditional statement is logically equivalent to its contrapositive.

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Let me know if you'd like a visual diagram or more examples!