Problem 8
Using \(\mathrm{T}\) to stand for a statement that is always true and \(\mathrm{F}\) to stand for a statement that is always false, give a simplified form of each of the following statements. a. \(\mathrm{T} \wedge \mathrm{s}\) b. \(F \wedge s\) c. \(\mathrm{T} \vee \mathrm{s}\) d. \(\mathrm{F} \vee \mathrm{s}\)
Problem 9
Rewrite the statement"The product of odd integers is odd" with all quantifiers (including any in the definition of odd integers) explicitly stated as "for all" or "there exist."
Problem 9
Prove that if \(x^{3}>8\), then \(x>2\).
Problem 10
Rewrite the following statement without any negations: "There is no positive integer \(n\) such that for all integers \(m>n\), all polynomial equations \(p(x)=0\) of degree \(m\) have no real numbers for solutions."
Problem 10
Give an example in English where "or" seems to mean "exclusive or" (or where you think it would for many people) and an example in English where "or" seems to mean "inclusive or" (or where you think it would for many people).
Problem 10
Prove that \(\sqrt{3}\) is irrational.
Problem 11
Construct a proof that if \(m\) is an integer such that \(m^{2}\) is even, then \(m\) is even.
Problem 11
Give an example in English where "if ... then" seems to mean "if and only if" (or where you think it would to many people) and an example in English where it seems not to mean "if and only if" (or where you think it would not to many people).
Problem 12
Find a statement involving only \(\wedge, \vee\), and \(\neg\) (and \(s\) and \(t\) ) equivalent to \(s \Leftrightarrow t\). Does your statement have as few symbols as possible? If you think it doesn't, try to find one with fewer symbols.
Problem 12
Prove or disprove the following statement: "For every positive integer \(n\), if \(n\) is prime, then 12 and \(n^{3}-n^{2}+n\) have a common factor greater than \(1 . "\)
