Problem 1
Write the converse and contrapositive of each statement. a. If the hose is \(60 \mathrm{ft}\) long, then the hose will reach the tomatoes. b. George goes for a walk only if Mary goes for a walk. c. Pamela recites a poem if Andre asked for a poem.
Problem 1
Give truth tables for the following expressions. a. \((s \vee t) \wedge(\neg s \vee t) \wedge(s \vee \neg t)\) b. \((s \Rightarrow t) \wedge(t \Rightarrow u)\) c. \((s \vee t \vee u) \wedge(s \vee \neg t \vee u)\)
Problem 1
For what positive integers \(x\) is the statement \((x-2)^{2}+1 \leq 2\) true? For what integers is it true? For what real numbers is it true? If you expand the universe for which you are considering a statement about a variable, does this always increase the size of the statement's truth set?
Problem 2
Is the statement "There is an integer greater than 2 such that \((x-2)^{2}+1 \leq 2\) " true or false? How do you know?
Problem 2
Construct a proof that if \(m\) is odd, then \(m^{2}\) is odd.
Problem 3
Construct a proof that for all integers \(m\) and \(n\), if \(m\) is even and \(n\) is odd, then \(m+n\) is odd.
Problem 3
Write the statement "The square of every real number is greater than or equal to \(0 "\) " as a quantified statement about the universe of real numbers. You may use \(R\) to stand for the universe of real numbers.
Problem 4
Show that the statements \(s \Rightarrow t\) and \(\neg s \vee t\) are equivalent.
Problem 4
A prime number is defined as an integer greater than 1 whose only positive integer factors are itself and 1. Find two ways to write this definition so that all quantifiers are explicit. (It may be convenient to introduce a variable to stand for the number and perhaps a variable or some variables for its factors.)
Problem 4
What does it really mean to say, "Prove that if \(m\) is odd, and \(n\) is odd, then \(m+n\) is even"? Prove this more precise statement.
