For each of the statements, begin a proof by contraposition and a proof by contradiction. This...

For each of the statements, begin a proof by contraposition and a proof by contradiction. This will include rewriting the statement, writing the assumptions, and writing what needs to be shown. From there, pick one of the two methods and finish the proof.

a) For all integers m and n, if m + n is even the m and n are both even or m and n are both odd.

b) For all integers a, b, and c, if a - bc then a - b. (Recall that the symbol - means “does not divide.”)

c) For all x ∈ Z, if x 2 − 6x + 5 is even, then x is odd.

2) Prove the following statement by contradiction: If a, b, and c are integers and a 2 + b 2 = c 2 , then at least one of a and b is even.

Expert Solution

Colby Messinger answered 1 year ago

3. To begin a proof by contradiction for “If n is even then n+1 is odd,”...

3. To begin a proof by contradiction for “If n is even then n+1 is odd,” what would you “assume true? 4. Prove that the following is not true by finding a counterexample. “The sum of any 3 consecutive integers is even" 5. Show a Proof by exhaustion for the following: For n = 2, 4, 6, n²-1 is odd 6. Show an informal Direct Proof for “The sum of 2 even integers is even.” Recursive Definitions 7. The Fibonacci Sequence is...

Define lemma? What is Proof by Contraposition? Define Proof?

Contradiction proof conception Prove: If A is true, then B is true Contradiction: If A is...

Contradiction proof conception Prove: If A is true, then B is true Contradiction: If A is true, then B is false. so we suppose B is false and follow the step to prove. At the end we get if A is true then B is true so contradict our assumption However, Theorem: Let (xn) be a sequence in R. Let L∈R. If every subsequence of (xn) has a further subsequence that converges to L, then (xn) converges to L. Proof: Assume,...

Prove by contraposition and again by contradiction: For all integers a,b, and c, if a divides...

Prove by contraposition and again by contradiction: For all integers a,b, and c, if a divides b and a does not divide c then a does not divide b + c Elaboration with definitions / properties used would be appreciated! Thanks in advance!!

Discrete Structures Use a proof by contraposition to show if x3 + 3x is an irrational...

Discrete Structures Use a proof by contraposition to show if x3 + 3x is an irrational number then so is x, for any real number x.

Exercise 2.4.1: Proofs by contradiction. About Give a proof for each statement. (c)The average of three...

Exercise 2.4.1: Proofs by contradiction. About Give a proof for each statement. (c)The average of three real numbers is greater than or equal to at least one of the numbers. (e)There is no smallest integer.

Suppose that a and b are integers. (a) (5 pts) Use a proof by contradiction to...

Suppose that a and b are integers. (a) (5 pts) Use a proof by contradiction to prove the statement ”If a − b is even, then a and b have the same parity (that is, they are both even or both odd).” Carefully show your algebra. (b) (5 pts) Show that any odd integer cubed is also an odd integer. Carefully show your algebra. (c) (5 pts) Use your results from parts (a-c) in a direct proof to show that...

Give a proof by contradiction to show that if two lines l and m are cut...

Give a proof by contradiction to show that if two lines l and m are cut by a transversal in such a way that the alternate interior angles, x and y, have the same measure, then the lines are parallel. Write the "if, then" statement for the proof by contradiction and the proof.

4. Use a proof by contradiction to show that the square root of 3 is irrational....

4. Use a proof by contradiction to show that the square root of 3 is irrational. You may use the following fact: For any integer k, if k2 is a multiple of 3, then k is a multiple of 3. Hint: The proof is very similar to the proof that √2 is irrational. 5. Use a direct proof to show that the product of a rational number and an integer must be a rational number. 6. Use a proof by...

Use proof by contradiction to show that a cycle graph with an odd number of nodes...

Use proof by contradiction to show that a cycle graph with an odd number of nodes is not 2-colorable.

Question