Introduction to Proof and Problem Solving
February 1, 2016
Problem 4. Prove or disprove: For every real number a > 0, there exists a real number b
such that for every x > b, 2/(3x + 5) < a.
Proof. Choose a = a0 > 0 to be any real
Let P(n), Q(n), and R(n) be three propositions.
Now consider P(n) P(n).
By definition of , this shows that is reflexive.
Now consider P(n) Q(n). By definition of , this implies that Q(n) P(n) and is symmetric.
1) If n is a positive integer and s is an irrational number, then n/s is an irrational
Scratch Work: Let P = P1 V P2, where P1 = n is a positive integer and P2 = s is an
irrational number. Let Q = n/
Exercise 1) Implications
Truth Table for Implications
a. Suppose P Q is true. Then the statement It is possible that Q
may be false is true. With reference to the truth table, there are 3
Find positive integers m and n such that m > 1, n >1 and mn and nm.
Let m = 2 and n = 4. Then m > 1 and n >1, and so, 24 = 42 = 16.
Three examples of the starting location for
February 12, 2014
I did not give or receive help on this exam.
1) Consider the statement
Write the contrapositive form of statement . DO NOT PROVE ANYTHING.
2) Write the negation of statement
INTRO TO PROOF AND PROBLEM SOLVING
1. Use the Principle of Mathematical Induction to prove the following for all natural numbers n .
(a) 3n 1 + 2n .
(b) 10n+1 + 3 4n1 + 5 is divisible by 9.
(c) 3n +3 > (n + 3)3 .
i =1 i
(e) For every p