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 𝑎 and 𝑏 are odd numbers, then 𝑎𝑏 is odd.

Statement 2: 𝑐 and 𝑑 are odd numbers.

What can we conclude about 𝑐𝑑?

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

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

Statement 4: 𝑞 is an odd number.

Can we conclude 𝑞 is prime? Why?

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

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 𝑎𝑏 is odd, then both 𝑎 and 𝑏 are odd.

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

  • AIf 𝑎𝑏 is odd, then 𝑎+𝑏 is even.
  • BIf 𝑎 is odd, then both 𝑎+𝑏 and 𝑎𝑏 are odd.
  • CIf 𝑎+𝑏 is even, then 𝑎𝑏 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 𝑄 has a right angle and a pair of parallel sides. What can we conclude about shape 𝑄 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: Daniel 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.
  • BDaniel is young.
  • CDaniel 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: 𝑝𝑞𝑞𝑝
  • BThe law of detachment: 𝑝𝑞𝑝𝑝
  • CThe law of syllogism: 𝑝𝑞𝑝𝑞

Q9:

If you wear sunglasses, then it is sunny outside.

It is cloudy (not sunny).

Write the symbolic representation.

  • AThe law of detachment: 𝑝𝑞¬𝑝¬𝑞
  • BThe law of contrapositive: 𝑝𝑞¬𝑞¬𝑝
  • CThe law of detachment: 𝑝𝑞¬𝑞¬𝑝

Q10:

All equiangular triangles are equilateral.

𝐴 𝐵 𝐶 is equiangular.

Which of the following is true?

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

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