Introduction to Proof: Direct and Indirect
Writing logical arguments using direct proof and proof by contradiction
Introduction to Proof: Direct and Indirect
Writing logical arguments using direct proof and proof by contradiction
Math - Grade 9-12
- 1
State the hypothesis and conclusion of this conditional statement: If a number is divisible by 6, then it is divisible by 3.
The hypothesis follows the word if, and the conclusion follows the word then.
The hypothesis is that a number is divisible by 6. The conclusion is that the number is divisible by 3. - 2
Write a direct proof of the statement: If n is an even integer, then n + 4 is even.
Assume n is an even integer. Then n = 2k for some integer k. So n + 4 = 2k + 4 = 2(k + 2). Since k + 2 is an integer, n + 4 is even. - 3
Write a direct proof of the statement: If a and b are odd integers, then a + b is even.
Use the definition of an odd integer.
Assume a and b are odd integers. Then a = 2m + 1 and b = 2n + 1 for some integers m and n. Their sum is a + b = 2m + 1 + 2n + 1 = 2m + 2n + 2 = 2(m + n + 1). Since m + n + 1 is an integer, a + b is even. - 4
A student begins a proof with this sentence: Assume x is an integer and x is divisible by 10. What type of proof is the student most likely writing if the goal is to prove x is divisible by 5?
The student is most likely writing a direct proof because the student begins by assuming the hypothesis and will use definitions to reach the conclusion. - 5
Write the first sentence of an indirect proof of this statement: If two lines are perpendicular, then they intersect at a right angle.
In an indirect proof, begin by assuming the opposite of the conclusion.
Assume two lines are perpendicular but they do not intersect at a right angle. This assumes the hypothesis is true and the conclusion is false. - 6
Use an indirect proof to prove: If n is an integer and n is odd, then n is not divisible by 2.
Assume n is an odd integer and suppose, for contradiction, that n is divisible by 2. Then n = 2k for some integer k, which means n is even. This contradicts the fact that n is odd. Therefore, n is not divisible by 2. - 7
Decide whether a direct proof or an indirect proof is more natural for this statement, and explain your choice: There is no smallest positive real number.
Try assuming the statement is false.
An indirect proof is more natural. We can assume there is a smallest positive real number r, then note that r/2 is also positive and smaller than r. This creates a contradiction. - 8
Complete the missing step in this direct proof: If n is divisible by 4, then n is even. Assume n is divisible by 4. Then n = 4k for some integer k. Since 4k = 2(____), n is even.
The missing expression is 2k. Since n = 4k = 2(2k), and 2k is an integer, n is even. - 9
Write the contrapositive of this statement: If a triangle is equilateral, then it is isosceles.
Switch the hypothesis and conclusion, then negate both parts.
The contrapositive is: If a triangle is not isosceles, then it is not equilateral. - 10
Prove the statement by proving its contrapositive: If n squared is even, then n is even.
The contrapositive is: If n is odd, then n squared is odd. Assume n is odd, so n = 2k + 1 for some integer k. Then n squared = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1. Since 2k^2 + 2k is an integer, n squared is odd. Therefore, if n squared is even, then n is even. - 11
Find the error in this proof: Claim: If n is even, then n + 1 is even. Proof: Assume n is even, so n = 2k. Then n + 1 = 2k + 1, which is even. Therefore, n + 1 is even.
Use the definitions of even and odd integers.
The error is the statement that 2k + 1 is even. An expression of the form 2k + 1 is odd, so the proof reaches a false conclusion. - 12
Use an indirect proof to prove: If two angles form a linear pair, then they cannot both be acute.
Assume two angles form a linear pair and suppose, for contradiction, that both angles are acute. Each acute angle has measure less than 90 degrees, so their sum is less than 180 degrees. But angles in a linear pair have measures that sum to 180 degrees. This is a contradiction, so the two angles cannot both be acute.