Section 1
Proof strategies
Consider the following theorem. (This theorem was proven in the introduction.)Suppose $n$ is an integer larger than 1 and $n$ is not prime. Then $2^n-1$ is not prime.(a) Identify the hypotheses and conclusion of the theorem. Are the hypotheses true when $n=6$ ? What does the theorem tell you in this instance? Is it right?(b) What can you conclude from the theorem in the case $n=15$ ? Check directly that this conclusion is correct.(c) What can you conclude from the theorem in the case $n=11$ ?
Consider the following incorrect theorem:Suppose $n$ is a natural number larger than 2, and $n$ is not a prime number. Then $2 n+13$ is not a prime number.What are the hypotheses and conclusion of this theorem? Show that the theorem is incorrect by finding a counterexample.
Complete the following alternative proof of the theorem in Example 3.1.2.Proof. Suppose $0<a<b$. Then $b-a>0$.[Fill in a proof of $b^2-a^2>0$ here.]Since $b^2-a^2>0$, it follows that $a^2<b^2$. Therefore if $0<a<b$ then $a^2<b^2$.
Suppose $A \backslash B \subseteq C \cap D$ and $x \in A$. Prove that if $x \notin D$ then $x \in B$.
Suppose $a$ and $b$ are real numbers. Prove that if $a<b$ then $\frac{a+b}{2}<b$.
Suppose $x$ is a real number and $x \neq 0$. Prove that if $\frac{\sqrt[3]{x}+5}{x^2+6}=\frac{1}{x}$ then $x \neq 8$.
Suppose $a, b, c$, and $d$ are real numbers, $0<a<b$, and $d>0$. Prove that if $a c \geq b d$ then $c>d$.
Suppose $x$ and $y$ are real numbers, and $3 x+2 y \geq 5$. Prove that if $x>1$ then $y<1$.
Prove the first theorem in Example 3.1.1. (Hint: You might find it useful to apply the theorem from Example 3.1.2.)
Consider the following theorem.Theorem. Suppose $x$ is a real number and $x \neq 4$. If $\frac{2 x-5}{x-4}=3$ then $x=7$.(a) What's wrong with the following proof of the theorem?Proof. Suppose $x=7$. Then $\frac{2 x-5}{x-4}=\frac{2(7)-5}{7-4}=\frac{9}{3}=3$. Therefore if $\frac{2 x-5}{x-4}=3$ then $x=7$.(b) Give a correct proof of the theorem.