Question

Using either proof by contraposition or proof by contradiction, show that: if n2 + n is...

  1. Using either proof by contraposition or proof by contradiction, show that:

    if n2 + n is irrational, then n is irrational.
  2. Using the definitions of odd and even show that the following 4 statements are equivalent:
    1. n2 is odd
    2. 1 − n is even
    3. n3 is odd
    4. n + 1 is even
Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Using either proof by contraposition or proof by contradiction, show that: if n2 + n is...
Using either proof by contraposition or proof by contradiction, show that: if n2 + n is irrational, then n is irrational.
Statement: "For all integers n, if n2 is odd then n is odd" (1) prove the...
Statement: "For all integers n, if n2 is odd then n is odd" (1) prove the statement using Proof by Contradiction (2) prove the statement using Proof by Contraposition
Prove by either contradiction or contraposition: For all integers m and n, if m+n is even...
Prove by either contradiction or contraposition: For all integers m and n, if m+n is even then m and n are either both even or both odd.
Hint: it is sufficient to show A implies B, B implies C, C implies D, and...
Hint: it is sufficient to show A implies B, B implies C, C implies D, and D implies A, as repeated application of the hypothetical syllogism will give you A iff B iff C iff D. Using the definitions of odd and even show that the following 4 statements are equivalent: n2 is odd 1 − n is even n3 is odd n + 1 is even
Definition of Even: An integer n ∈ Z is even if there exists an integer q...
Definition of Even: An integer n ∈ Z is even if there exists an integer q ∈ Z such that n = 2q. Definition of Odd: An integer n ∈ Z is odd if there exists an integer q ∈ Z such that n = 2q + 1. Use these definitions to prove the following: Prove that zero is not odd. (Proof by contradiction)
Ex 2. Prove by contradiction the following claims. In each proof highlight what is the contradiction...
Ex 2. Prove by contradiction the following claims. In each proof highlight what is the contradiction (i.e. identify the proposition Q such that you have Q ∧ (∼Q)). Claim 1: The sum of a rational number and an irrational number is irrational. (Recall that x is said to be a rational number if there exist integers a and b, with b 6= 0 such that x = a b ). Claim 2: There is no smallest rational number strictly greater...
Prove or disprove the following statements. Remember to disprove a statement you have to show that...
Prove or disprove the following statements. Remember to disprove a statement you have to show that the statement is false. Equivalently, you can prove that the negation of the statement is true. Clearly state it, if a statement is True or False. In your proof, you can use ”obvious facts” and simple theorems that we have proved previously in lecture. (a) For all real numbers x and y, “if x and y are irrational, then x+y is irrational”. (b) For...
Theorem: If m is an even number and n is an odd number, then m^2+n^2+1 is...
Theorem: If m is an even number and n is an odd number, then m^2+n^2+1 is even. Don’t prove it. In writing a proof by contraposition, what is your “Given” (assumption)? ___________________________ What is “To Prove”: _____________________________
Show that if n2+2 and n2-2 are both prime, then 3 divides n. Note: This question...
Show that if n2+2 and n2-2 are both prime, then 3 divides n. Note: This question was asked in the context of congruences/ modulus in a Number Theory class. Please give a NEATLY written proof in full sentences. State any theorems, definitions, and formulas used.
1. For each statement that is true, give a proof and for each false statement, give...
1. For each statement that is true, give a proof and for each false statement, give a counterexample     (a) For all natural numbers n, n2 +n + 17 is prime.     (b) p Þ q and ~ p Þ ~ q are NOT logically equivalent.     (c) For every real number x ³ 1, x2£ x3.     (d) No rational number x satisfies x^4+ 1/x -(x+1)^(1/2)=0.     (e) There do not exist irrational numbers x and y such that...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT