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 contradiction to show that the sum of an integer and an irrational number must be irrational.

Expert Solution

Colby Messinger answered 2 years ago

Prove that √3 is irrational using contradiction. You can use problem 4 as a lemma for...

Prove that √3 is irrational using contradiction. You can use problem 4 as a lemma for this. Problem 4, for context is Prove that if n2 is divisible by 3, then n is divisible by 3.

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.

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.

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

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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...

Prove by contradiction, every real solution of x3+x+3=0 is irrational.

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...

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

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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...

Show that for any square-free integer n > 1, √ n is an irrational number

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