91Ó°ÊÓ

Construct a know-show table for each of the following statements and then write a formal proof for one of the statements. (a) If \(m\) is an even integer, then \(m+1\) is an odd integer. (b) If \(m\) is an odd integer, then \(m+1\) is an even integer.

Construct a know-show table for each of the following statements and then write a formal proof for one of the statements. (a) If \(x\) is an even integer and \(y\) is an even integer, then \(x+y\) is an even integer. (b) If \(x\) is an even integer and \(y\) is an odd integer, then \(x+y\) is an odd integer. (c) If \(x\) is an odd integer and \(y\) is an odd integer, then \(x+y\) is an even integer.

Identify the hypothesis and the conclusion for each of the following conditional statements. (a) If \(n\) is a prime number, then \(n^{2}\) has three positive factors. (b) If \(a\) is an irrational number and \(b\) is an irrational number, then \(a \cdot b\) is an irrational number. (c) If \(p\) is a prime number, then \(p=2\) or \(p\) is an odd number. (d) If \(p\) is a prime number and \(p \neq 2,\) then \(p\) is an odd number. (e) If \(p \neq 2\) and \(p\) is an even number, then \(p\) is not prime.

Determine whether each of the following conditional statements is true or false. (a) If \(10<7,\) then \(3=4\). (c) If \(10<7,\) then \(3+5=8\). (b) If \(7<10,\) then \(3=4\). (d) If \(7<10,\) then \(3+5=8\).

Construct a know-show table and write a complete proof for each of the following statements: (a) If \(m\) is an even integer, then \(3 m^{2}+2 m+3\) is an odd integer. (b) If \(m\) is an odd integer, then \(3 m^{2}+7 m+12\) is an even integer.

Let \(P\) be the statement "Student X passed every assignment in Calculus I" and let \(Q\) be the statement "Student \(X\) received a grade of \(C\) or better in Calculus I." (a) What does it mean for \(P\) to be true? What does it mean for \(Q\) to be true? (b) Suppose that Student X passed every assignment in Calculus I and received a grade of \(\mathrm{B}-,\) and that the instructor made the statement \(P \rightarrow Q\). Would you say that the instructor lied or told the truth? (c) Suppose that Student X passed every assignment in Calculus I and received a grade of \(\mathrm{C}-,\) and that the instructor made the statement \(P \rightarrow Q\). Would you say that the instructor lied or told the truth? (d) Now suppose that Student \(X\) did not pass two assignments in Calculus I and received a grade of \(D\), and that the instructor made the statement \(P \rightarrow Q\). Would you say that the instructor lied or told the truth? (e) How are Parts (5b), (5c), and (5d) related to the truth table for \(P \rightarrow Q\) ?

Are the following statements true or false? Justify your conclusions. (a) If \(a, b\) and \(c\) are integers, then \(a b+a c\) is an even integer. (b) If \(b\) and \(c\) are odd integers and \(a\) is an integer, then \(a b+a c\) is an even integer.

Following is a statement of a theorem which can be proven using the quadratic formula. For this theorem, \(a, b,\) and \(c\) are real numbers. Theorem If \(f\) is a quadratic function of the form \(f(x)=a x^{2}+b x+c\) and \(a c<0,\) then the function \(f\) has two \(x\) -intercepts. Using only this theorem, what can be concluded about the functions given by the following formulas? (a) \(g(x)=-8 x^{2}+5 x-2\) (b) \(h(x)=-\frac{1}{3} x^{2}+3 x\) (c) \(k(x)=8 x^{2}-5 x-7\) (d) \(j(x)=-\frac{71}{99} x^{2}+210\) (e) \(f(x)=-4 x^{2}-3 x+7\) (f) \(F(x)=-x^{4}+x^{3}+9\)

Is the following statement true or false? Justify your conclusion. If \(a\) and \(b\) are nonnegative real numbers and \(a+b=0,\) then \(a=0\). Either give a counterexample to show that it is false or outline a proof by completing a know-show table.

Following is a statement of a theorem about certain cubic equations. For this theorem, \(b\) represents a real number. Theorem \(\mathbf{A}\). If \(f\) is a cubic function of the form \(f(x)=x^{3}-x+b\) and \(b>1,\) then the function \(f\) has exactly one \(x\) -intercept. Following is another theorem about \(x\) -intercepts of functions: Theorem \(\mathbf{B}\). If \(f\) and \(g\) are functions with \(g(x)=k \cdot f(x),\) where \(k\) is a nonzero real number, then \(f\) and \(g\) have exactly the same \(x\) -intercepts. Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas? (a) \(f(x)=x^{3}-x+7\) (b) \(g(x)=x^{3}+x+7\) (c) \(h(x)=-x^{3}+x-5\) (d) \(k(x)=2 x^{3}+2 x+3\) (e) \(r(x)=x^{4}-x+11\) (f) \(F(x)=2 x^{3}-2 x+7\)