Question

1. \if a shape has exactly three sides, then the shape is a triangle.\ converse: if a shape is a triangle ,then the shape has exactly three sides . inverse: if a shape select , then the shape select . contrapositive: if a shape select , then the select .

Explanation:

To solve the problem of finding the inverse and contrapositive of the given conditional statement, we start by recalling the definitions of these logical forms.

Step 1: Identify the original statement

The original statement is: "If a shape has exactly three sides, then the shape is a triangle."
Let \( p \) represent "a shape has exactly three sides" and \( q \) represent "the shape is a triangle." So the original statement is \( p
ightarrow q \).

Step 2: Find the Inverse

The inverse of a conditional statement \( p
ightarrow q \) is \(
eg p
ightarrow
eg q \), where \(
eg \) denotes "not."

• \(

eg p \): "a shape does not have exactly three sides"

• \(

eg q \): "the shape is not a triangle"

So the inverse is: "If a shape does not have exactly three sides, then the shape is not a triangle."

Step 3: Find the Contrapositive

The contrapositive of a conditional statement \( p
ightarrow q \) is \(
eg q
ightarrow
eg p \).

• \(

eg q \): "a shape is not a triangle"

• \(

eg p \): "the shape does not have exactly three sides"

So the contrapositive is: "If a shape is not a triangle, then the shape does not have exactly three sides."

Final Answers:

• Inverse: If a shape \(\boldsymbol{\text{does not have exactly three sides}}\), then the shape \(\boldsymbol{\text{is not a triangle}}\).
• Contrapositive: If a shape \(\boldsymbol{\text{is not a triangle}}\), then the shape \(\boldsymbol{\text{does not have exactly three sides}}\).

Answer:

To solve the problem of finding the inverse and contrapositive of the given conditional statement, we start by recalling the definitions of these logical forms.

Step 1: Identify the original statement

The original statement is: "If a shape has exactly three sides, then the shape is a triangle."
Let \( p \) represent "a shape has exactly three sides" and \( q \) represent "the shape is a triangle." So the original statement is \( p
ightarrow q \).

Step 2: Find the Inverse

The inverse of a conditional statement \( p
ightarrow q \) is \(
eg p
ightarrow
eg q \), where \(
eg \) denotes "not."

• \(

eg p \): "a shape does not have exactly three sides"

• \(

eg q \): "the shape is not a triangle"

So the inverse is: "If a shape does not have exactly three sides, then the shape is not a triangle."

Step 3: Find the Contrapositive

The contrapositive of a conditional statement \( p
ightarrow q \) is \(
eg q
ightarrow
eg p \).

• \(

eg q \): "a shape is not a triangle"

• \(

eg p \): "the shape does not have exactly three sides"

So the contrapositive is: "If a shape is not a triangle, then the shape does not have exactly three sides."

Final Answers:

• Inverse: If a shape \(\boldsymbol{\text{does not have exactly three sides}}\), then the shape \(\boldsymbol{\text{is not a triangle}}\).
• Contrapositive: If a shape \(\boldsymbol{\text{is not a triangle}}\), then the shape \(\boldsymbol{\text{does not have exactly three sides}}\).