# Lesson Worksheet: Proof by Contradiction Mathematics

In this worksheet, we will practice describing what proof by contradiction is and using it to determine whether a conjecture is true or false.

Q1:

A student has tried to prove the following statement by contradiction: “For all even integers and , their product is divisible by 4.” The student’s working is shown below.

1. Assumption: All integers such that is odd are even.
2. is even; so, write
3. Then, , so is even.
4. This contradicts the assumption that is odd.
5. Therefore, if is odd, then must be odd.

Identify the error in their working.

• AIn the second sentence, they should have written and .
• BIn the final sentence, they have drawn the wrong conclusion.
• CIn the fourth sentence, they have stated the wrong assumption.
• DIn the third sentence, the algebra is incorrect.
• EIn the first sentence, they have given the incorrect negation of the statement they want to prove.

Q2:

Choose the statement that is the negation of “all multiples of 5 are odd.”

• AAll odd numbers are multiples of 5.
• BNo multiple of 5 is odd.
• CAll multiples of 5 are even.
• DAt least one multiple of 5 is even.
• EMore than one multiple of 5 is even.

Q3:

What is the negation of each of the following statements?

All swans are white.

• AAt least one swan is not white.
• BMore than one swan is not white.
• CNo swans are white.
• DAll swans are not white.
• EAll swans are black.

If and are prime numbers, then is a prime number.

• AThere exist prime numbers and such that is not prime.
• BIf is a prime number, then and are prime numbers.
• CIf and are not prime numbers, then there exists a number of the form that is prime.
• DIf and are not prime numbers, then is not prime.
• EIf and are prime numbers, then is not prime.

All odd numbers are either multiples of 3 or can be written in the form .

• AThere is an odd number that is neither a multiple of 3 nor can be written in the form .
• BAll numbers that are either multiples of 3 or can be written in the form are odd.
• CNo numbers that are either multiples of 3 or can be written in the form are odd.
• DAll even numbers are either multiples of 3 or can be written in the form .
• ENo odd numbers are either multiples of 3 or can be written in the form .

Q4:

Madison wants to prove by contradiction that if is an irrational number, then at least one of and is irrational.

Which of the following assumptions should Madison use at the start of her proof?

• AAssume that is an irrational number and that neither nor is irrational.
• BAssume that is an irrational number and that both and are irrational.
• CAssume that is a rational number and that neither nor is irrational.
• DAssume that is a rational number and that at least one of and is irrational.
• EAssume that at least one of and is an irrational number and that is irrational.

Which of the following statements is the contradiction that Madison should obtain near the end of her proof?

• AAssuming that neither nor is irrational leads to the contradiction that is rational.
• BAssuming that neither nor is irrational leads to the contradiction that is irrational.
• CAssuming that or is irrational leads to the contradiction that is irrational.
• DAssuming that both and are irrational leads to the contradiction that is rational.
• EAssuming that is irrational leads to the contradiction that neither nor is irrational.

Madison’s classmate makes this statement: “If is a rational number, then at least one of and must be rational.”

Disprove the statement by choosing a suitable counterexample from the following.

• A and
• B and
• C and
• D and
• E and