Show all your work; answers alone will receive no credit. Use proper notation, s
ID: 3198627 • Letter: S
Question
Show all your work; answers alone will receive no credit. Use proper notation, standard conventions, and technical terminology 2) Rewrite the proposed theorem in the form "if (p) then (g)", label the hypothesis and the conclusion, identify the universe, then prove the proposed theorem a) 3n2 +21 is divisible by 12 whenever n is odd. b) If the "Best of the Beatles" musical anthology includes 53 songs on 5 CDs, then some CD must have at least 11 songs. C- Prove the theorem that you did not do in question 2.Explanation / Answer
Note : The theorem in question 2 is not mentioned. Giving answers to a) and b).
a) 3n2 + 21 is divisible by 12 whenever n is odd.
This statement can be written as:
If n is odd, 3n2 + 21 is divisible by 12.
The hypothesis is "n is odd" and the conclusion is "3n2 + 21 is divisible by 12".
The universe is the set of integers.
Proof : Since n is odd, let n = 2k + 1 where k is an integer.
=> 3n2 + 21 = 3 * (2k + 1)2 + 21
= 3 * (4k2 + 4k + 1) + 21
= 12k2 + 12k + 3 + 21
= 12k2 + 12k + 24
= 12 (k2 + k + 2)
Since there is a factor 12, 3n2 + 21 is divisible by 12.
b) If the "Best of Beatles" musical anthology includes 53 songs on 5 CDs, then some CD must have atleast 11 songs.
The hypothesis is "The 'Best of Beatles' musical anthology includes 53 songs on 5 CDs" and the conclusion is "some CD must have atleast 11 songs".
The universe is the songs of Beatles.
Proof : We have by the Pigeon-hole principle taking songs as pigeons and CDs as holes
=> There should be atleast one CD that contains [53/5] + 1 songs.
=> There should be atleast one CD that contains 10 + 1 songs.
=> There should be some CD that contains 11 songs.