Discrete Mathematics : Ten Proofs

Prove the given statement: 1) The sum of an integer and it's square is even. 2) The sum of the squares of two odd integers cannot be a perfect square. 3) The sum of any three consecutive integers is even. 4) The product of two rational numbers is rational. 5) The product of two irrational numbers is irrational. ...continues

Complete Graphs and Cycles; Undirected & Spanning Trees and Reverse Polish Notation

Consider a comple graph G, n ≥ 3. Find the number of cycles in G of length n. How many cycles in a complete graph with 5 vertices? Another problem is attached involving Reverse Polish Notation.... Please see the attached file for the fully formatted problems.

Proofs by induction

I'm having a hard time comprehending how to write proofs by induction. I'm looking for answers to these problems so that I may have a better understanding of how they are done. --- All problems need to be proved using induction in proofs. 1. Consider n infinitely long straight lines, none of which are parallel and no th ...continues

Prove or disprove that if a and b are rational numbers the a^b (a to the power b ) is also rational.

Prove or disprove that if a and b are rational numbers the a^b (a to the power b ) is also rational.

Proof - show that if r is an irrational number, there is a unique integer n such that the distance between r and n is less than 1/2.

Show that if r is an irrational number, there is a unique integer n such that the distance between r and n is less than 1/2.

Prove that if n is a positive integer then n is even if and only if 7n+4 is even.

Prove that if n is a positive integer then n is even if and only if 7n+4 is even.

Perfect Square Proof

Prove that 2 * 10^500 + 15 or 2 * 10^500 + 16 is not a perfect square. Is your proof constructive or nonconstructive? Note: ^ is to the power of

Prove that these four statements are equivalent: i)n^2 is odd, (ii) 1 - n is even, (iii) n^3 is odd , (iv) n^2 +1 is even

Prove that these four statements are equivalent: i) n^2 is odd, (ii) 1 - n is even, (iii) n^3 is odd , (iv) n^2 +1 is even

Methods of proof

Prove that the square of an even number is an even number using: a) a direct proof b) an indirect proof c) proof by contradiction

Methods of proof

Prove that if n is a positive integer, then n is even if and only if 7n+4 is even