# Worksheet: Laws of Deductive Reasoning

In this worksheet, we will practice using the law of detachment and the law of syllogism to reach logical conclusions from given statements.

**Q1: **

Consider the following two statements.

Statement 1: If A is true, then B is true.

Statement 2: A is true.

If both statements are true, then the law of detachment allows us to conclude that B is true.

Statement 1: If and are odd numbers, then is odd.

Statement 2: and are odd numbers.

What can we conclude about ?

- AIt is even.
- BWe cannot conclude anything.
- CIt is odd.

Statement 3: If is an odd number, then is prime.

Statement 4: is an odd number.

Can we conclude is prime? Why?

- ANo, because statement 3 is not a true statement.
- BYes, it follows from the law of detachment.
- CNo, because is not .

**Q2: **

Consider the following two statements:

Statement 1: If A is true, then B is true.

Statement 2: If B is true, then C is true.

If both statements are true, then the law of syllogism tells us that we can write a third true statement:

Statement 3: If A is true, then C is true.

Given statements 3 and 4 below, which of the following statements is also true?

Statement 3: If is odd, then both and are odd.

Statement 4: If both and are odd, then is even.

- AIf is odd, then is even.
- BIf is odd, then both and are odd.
- CIf is even, then is odd.

Statement 5: If a shape is a square, then it has a right angle.

Statement 6: If a shape is a square, then it has a pair of parallel sides.

Shape has a right angle and a pair of parallel sides. What can we conclude about shape from the given statements?

- AWe cannot conclude anything.
- BIt is a square.

**Q3: **

If you wear sunglasses, then it is sunny outside.

You are wearing sunglasses.

Which of the following is true?

- AIt is sunny outside, according to the law of syllogism.
- BIt is sunny outside, according to the law of detachment.
- CNothing can be concluded.

**Q4: **

Determine the conclusion from the true statements below.

Babies wear diapers.

My little brother does **not** wear diapers.

- ANothing can be concluded from these statements.
- BMy little brother is a baby.
- CMy little brother is not a baby.

**Q5: **

Consider the following premises.

Premise 1: Ethan is a primary school pupil.

Premise 2: All primary school pupils are young.

Which of the following conclusions is true?

- ANothing can be concluded from these premises.
- BEthan is young.
- CEthan is not young.

**Q6: **

If you text while driving, then you are unsafe.

You are a safe driver.

Which of the following is true?

- AYou do not text while driving, according to the law of contrapositive.
- BYou text while driving, according to the law of contrapositive.
- CYou do not text while driving, according to the law of deductive reasoning.

**Q7: **

If you are **not** in Giza, then you **cannot** be at the pyramids.

Matthew is at the pyramids.

Which of the following can be concluded from the above true statements?

- AMatthew is in Giza.
- BNothing can be concluded.
- CMatthew is not in Giza.

**Q8: **

If you wear sunglasses, then it is sunny outside.

It is sunny.

Write the symbolic representation.

- AThe law of detachment:
- BThe law of detachment:
- CThe law of syllogism:

**Q9: **

If you wear sunglasses, then it is sunny outside.

It is cloudy (not sunny).

Write the symbolic representation.

- AThe law of detachment:
- BThe law of contrapositive:
- CThe law of detachment:

**Q10: **

All equiangular triangles are equilateral.

is equiangular.

Which of the following is true?

- ANothing can be concluded.
- B is not equilateral, according to the law of detachment.
- C is equilateral, according to the law of detachment.