1. Prove that the difference of two odd integers is even. Give a justification a
ID: 3596718 • Letter: 1
Question
1. Prove that the difference of two odd integers is even. Give a justification at each step. (20 points)
2. Prove that the sum of any two rational numbers is a rational number. Give a justification at each step. (20 points)
3. Prove by contradiction that there is no greatest integer. (20 points)
4. Prove by contraposition that for all integers n, if n 2 is even then n is even. (20 points)
5. Prove using mathematical induction: integers n 1, 2 + 4 + ... + 2n = n 2 + n. Give justifications for each step. (20 points)
Explanation / Answer
1) This statement is true
By the definition even any integer multiple with 2 will be even, Let a = 2n+1 and b = 2m+1 be two arbitrary odd integers then
b - a = 2m + 1 - (2n + 1) = 2m - 2n = 2 * (m - n ) let m-n is k then 2(m-n)=2k
hence, Any integer multiplied by 2 gives even number. Here it proves that the difference of any two odd integers is even
2)This statement is true.
Let us assume m and n are two rational numbers.By considering the rational definition
m=a/b and n=c/d here a,b,c,d are integers and b,d != 0
Thus,consider m+n=a/b + c/d = ad+bc/bd where bd!=0 is an integer and ad+bc is an integer.
Then it specifies that the sum is rational by definition.
3)The statement is true
For contradiction suppose the statement to be proved false there fore negation of the statement is true.
Like wise suppose there is gretest integer N such that for all integers n.
where n<=N and consider N+1 here N and 1 are both integers then N+1 is an integer.
N+1>N this shows that N is not the greatest integer.
We supposed that N is the greatest integer and derived that N is not a greatest integer which is a contradiction.(i.e P/~P)
This shows assumption that there is a greatest integer is false,hence it proves that there is no greatest integer.
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