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.

Expert Solution

milcah answered 2 years ago

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.

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...

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,...

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...

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...

Let l and m be two lines in ordinary Euclidean geometry intersecting at point O. Let...

Let l and m be two lines in ordinary Euclidean geometry intersecting at point O. Let A, B and C be three distinct points on l and A', B' and C' three distinct points on m (none of them equal to O). Suppose that AB' is parallel to BA', and AC' is parallel to CA'. Prove, using Pappus' theorem in the Extended Euclidean Plane, that BC' and CB' are parallel.

1. Must be nicely written up AS A PROOF. a. Show that gcd(m + n, m)...

1. Must be nicely written up AS A PROOF. a. Show that gcd(m + n, m) = gcd(m, n). b. If n | k(n + 1), show that n | k. c. Show that any two consecutive odd integers are relatively prime.

The Alternate Interior Theorem states that if two lines cut by a transversal line have a...

The Alternate Interior Theorem states that if two lines cut by a transversal line have a pair of congruent alternate interior angles, then they must be parallel. Prove this theorem in Neutral Geometry (first 4 axioms) in two ways 1) using the exterior angle theorem 2) then without. Hint: use proof by contradiction for both solutions.

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