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. 6) Prove that the cube root of 2 is not a rational number. 7) Prove that the square root of 3 is not a rational number. 8) If n is an even prime number, then n=2. 9) If n is an even integer, 4 is less than or equal to n which is less than or equal to 12, then n is a sum of two prime numbers. 10) For every integer n, the number 3(n^2 + 2n + 3) - 2n^2 is a... click for more

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 three of which have a common point of intersection. Show that for n >= 1, the lines divide the plane into (n^2 + n + 2)/2 separate regions. 2. A string of 0s and 1s is to be processed and converted to an even-parity string by adding a parity bit to the end of the string. The parity bit is initially 0. When a 0 character is processed, the parity bit remains unchang... click for more

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.