Activity: Prove the following mathematical statements: Write proofs for the given statements, (inserting parenthetic remarks to explain the rationale behind each step) a) The product of an even number and any other number is even. (use direct proof) b) If n>0 and 4n−1 is prime, then n is odd. (use contradiction or indirect proof) c) The sum of the first n numbers is n(n+1)/2. (use mathematical induction)
Activity:
Prove the following mathematical statements:
Write proofs for the given statements, (inserting parenthetic remarks to explain the rationale behind each step)
a) The product of an even number and any other number is even. (use direct proof)
b) If n>0 and 4n−1 is prime, then n is odd. (use contradiction or indirect proof)
c) The sum of the first n numbers is n(n+1)/2. (use mathematical induction)
Example for Direct Proof:
1. If n is even, so is n2
a. Assume n is an even number, so n = 2k ; n2 = (2k)2
b. Thus n2 = 4k2 =2(2k2); Let j = 2k2
c. Then n2 = 2j
d. By definition n is even
Example for Proof by Contradiction:
1.For all real number, x and y,
if x + y ≥ 2, then x ≥ 1 and y ≥ 1.
a. Suppose the conclusion is false, that x < 1 and y < 1
b. So, (x < 1 ) + (y < 1) = x + y < 1 + 1 = 2
c. So, if x + y ≥ 2 and x + y = 2, then we come up with the realization that our claim is true
Example for Mathematical Induction:
1. 3n – 1 is a multiple of 2 for n = 1, 2, ...
Solution
Step 1. For n = 1 , 31 – 1 = 2 (a multiple of 2)
32 – 1 = 8
Step 2. Let us assume 3n -1 is true for n = k, Hence 3k – 1 is true (it is an assumption)
We have to prove that 3k+1 – 1 is also a multiple of 2
3k+1−1 = 3 × 3k−1 = (2 × 3k) + (3k − 1)
The first part ((2 x 3k) is certain to be a multiple of 2 and the second part (3k – 1) is also true as our previous assumption.
Hence, 3k +1 – 1 is a multiple of 2
So it is proved that 3n – 1 is a multiple of 2
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