How to Understand ‘If-Then’ Conditional Statements: A Comprehensive Guide

Step-by-step Guide: Conditional Statements

Defining Conditional Statements:
A conditional statement is a logical statement that has two parts: a hypothesis (the ‘if’ part) and a conclusion (the ‘then’ part). Written symbolically, it takes the form:
\( \text{If } p, \text{ then } q \)
Where \( p \) is the hypothesis and \( q \) is the conclusion.

Truth Values:
A conditional statement is either true or false. The only time a conditional statement is false is when the hypothesis is true, but the conclusion is false.

Converse, Inverse, and Contrapositive:
1. Converse: The converse of a conditional statement switches the hypothesis and the conclusion. For the statement “If \( p \), then \( q \)”, the converse is “If \( q \), then \( p \)”.

2. Inverse: The inverse of a conditional statement negates both the hypothesis and the conclusion. For the statement “If \( p \), then \( q \)”, the inverse is “If not \( p \), then not \( q \)”.

3. Contrapositive: The contrapositive of a conditional statement switches and negates both the hypothesis and the conclusion. For the statement “If \( p \), then \( q \)”, the contrapositive is “If not \( q \), then not \( p \)”.

Examples

Example 1: Simple Conditional
Statement: “If it is raining, then the ground is wet.”

Solution:
Hypothesis \(( p )\): It is raining.
Conclusion \(( q )\): The ground is wet.

Example 2: Determining Truth Value
Statement: “If a shape has four sides, then it is a rectangle.”

Solution:
This statement is false because a shape with four sides could be a square, trapezoid, or other quadrilateral, not necessarily a rectangle.

Example 3: Converse, Inverse, and Contrapositive
Statement: “If a number is even, then it is divisible by \(2\).”

Solution:
Converse: If a number is divisible by \(2\), then it is even.
Inverse: If a number is not even, then it is not divisible by \(2\).
Contrapositive: If a number is not divisible by \(2\), then it is not even.

Practice Questions:

1. Write the converse, inverse, and contrapositive for the statement: “If a bird is a penguin, then it cannot fly.”
2. Determine the truth value of the statement: “If a shape has three sides, then it is a triangle.”
3. For the statement “If an animal is a cat, then it is a mammal,” which of the following is its converse?
   a) If an animal is a mammal, then it is a cat.
   b) If an animal is not a cat, then it is not a mammal.
   c) If an animal is not a mammal, then it is not a cat.

Answers:

1. Converse: If a bird cannot fly, then it is a penguin.
   Inverse: If a bird is not a penguin, then it can fly.
   Contrapositive: If a bird can fly, then it is not a penguin.
2. The statement is true. A shape with three sides is defined as a triangle.
3. a) If an animal is a mammal, then it is a cat.