Question

what is the contrapositive of the following statement?

\if there is rain, then the dog will not bark.\

• if the dog will bark, then there is no rain.
• if there is no rain, then the dog will not bark.
• if the dog will not bark, then there is rain.
• if there is no rain, then the dog will bark.

Explanation:

Identify the hypothesis and conclusion

The original statement is a conditional statement.
Using Conditional Logic, we identify the components:

• Hypothesis \(p\): "there is rain"
• Conclusion \(q\): "the dog will not bark"

Define the contrapositive operation

The contrapositive of a conditional statement \(p
ightarrow q\) is \(
eg q
ightarrow
eg p\).
This means we negate both parts and swap their order.

Negate the components

• Negation of conclusion \(

eg q\): "the dog will bark"

• Negation of hypothesis \(

eg p\): "there is no rain"

Formulate the contrapositive statement

Combine the negated components into an "If..., then..." structure:
"If the dog will bark, then there is no rain."

Match with the given options

Compare our formulated statement with the choices:

• Option 1: "If the dog will bark, then there is no rain." (Matches)
• Option 2: "If there is no rain, then the dog will not bark." (Inverse)
• Option 3: "If the dog will not bark, then there is rain." (Converse)
• Option 4: "If there is no rain, then the dog will bark." (Negated converse)

Answer:

• (A) If the dog will bark, then there is no rain. (Correct answer)
• (B) If there is no rain, then the dog will not bark.
• (C) If the dog will not bark, then there is rain.
• (D) If there is no rain, then the dog will bark.