Chapter 1.5, Problem 7E

(a)

To determine

To prove: $ac$ divides $bc$ if and only if a divides b .

Explanation of Solution

Given information: Suppose a , b , c and d are positive integers.

Proof:

To prove $P\iff Q\text{ prove that }q\Rightarrow P\text{ and }P\Rightarrow Q$

  $\begin{array}{l}P=ac\text{ divides }bc\\ Q=a\text{ divides }b\end{array}$

  $\begin{array}{l}P\Rightarrow Q\\ P\\ \Updownarrow \\ \left(\exists q\right)\left(acq=bc\right)\\ \Updownarrow \\ \left(\exists q\right)\left(aq=b\right)\\ \Updownarrow \\ Q\\ P\Leftarrow Q\end{array}$

(b)

To determine

To prove: a + 1 divides b and b divides b + 3 if and only if a = 2 and b = 3.

Explanation of Solution

Given information: Suppose a , b , c and d are positive integers.

Proof:

To prove $P\iff Q\text{ prove that }q\Rightarrow P\text{ and }P\Rightarrow Q$

  $\begin{array}{l}P=a+b\text{ divides }b,b\text{ divides }b+3\\ Q=a=2\text{ and }b=3\end{array}$

  $\begin{array}{l}P\Rightarrow Q\\ P\\ \Updownarrow \\ \left(\exists q,p\right)\left(\left(a+1\right)q=b\right)\wedge \left(bp=b+3\right)\\ \Updownarrow \\ b\left(p-1\right)=3\Rightarrow b=3\vee b=1\\ \Updownarrow \\ \left(a+1\right)q=3\vee \left(a+1\right)q=1\\ \Updownarrow \\ a+1=3\vee a+1=1\\ \Updownarrow \\ a=2\vee a=0\\ \Updownarrow \\ Q\\ P\Leftarrow Q\end{array}$

(c)

To determine

To prove: a + c and $2b-a=d$ if and only if a = b − c and b + c = d .

Explanation of Solution

Given information: Suppose a , b , c and d are positive integers.

Proof:

To prove $P\iff Q\text{ prove that }q\Rightarrow P\text{ and }P\Rightarrow Q$

  $\begin{array}{l}P=a+c\text{ and 2}b-a=d\\ Q=a=b-c\text{ and }b+c=d\end{array}$

  $\begin{array}{l}P\Rightarrow Q\\ P\\ \Updownarrow \\ \text{The sum of first two equations: }a+c+2b-a=b+d\\ \text{rearranging the first equation: }a=b-c\\ \Updownarrow \\ \text{the first equation: }c+b=d\\ \Updownarrow \\ Q\\ P\Leftarrow Q\end{array}$

(d)

To determine

To prove: $a+2c\ne d\text{ or }b-a\ne 2d$ if and only if $b+2c\ne 3d$ or $3a+4c\ne b$ .

Explanation of Solution

Given information: Suppose a , b , c and d are positive integers.

To prove $P\iff Q\text{ prove that }q\Rightarrow P\text{ and }P\Rightarrow Q$

  $\begin{array}{l}P=a+2c\ne d\text{ and }b-a\ne 2d\\ Q=b+2c\ne 3d\text{ and 3}a+4c\ne b\end{array}$

  $\begin{array}{l}P\Rightarrow Q\\ P\\ \Updownarrow \\ \text{The sum of first two equations: }a+2c+b-a=d+2d\\ \text{multiply first elution by 2 and subtract the}\\ \text{second equation: 2}a+4c-b+a\ne 2d-2d\\ \Updownarrow \\ 2c+b=3d\\ 3a+4c\ne d\\ \Updownarrow \\ Q\\ P\Leftarrow Q\end{array}$

(e)

To determine

To prove: $a^2,a+b\text{ and }a+b+c$ are all odd if and only if $ab+bc$ and $b+c$ are even and

a + c is odd.

Explanation of Solution

Given information: Suppose a , b , c and d are positive integers.

Proof:

To prove $P\iff Q\text{ prove that }q\Rightarrow P\text{ and }P\Rightarrow Q$

  $\begin{array}{l}P=a^2,a+b,a+b+c\text{ are odd}\\ Q=ab+bc,b+c\text{ are even and }a+c\text{ are odd}\end{array}$

  $\begin{array}{l}P\Rightarrow Q\\ P\\ \Updownarrow \text{ 5}\left(\text{g}\right)\\ a\text{ is odd}\\ \Updownarrow \text{ 5}\left(\text{h}\right)\\ b\text{ is even and }c\text{ is even}\\ \Updownarrow \text{ 5}\left(\text{h}\right)\\ b+c\text{ is even and }a+c\text{ is odd}\\ \Updownarrow \text{ 5}\left(\text{b}\right)\\ ab\text{ is even and }bc\text{ is even}\\ \Updownarrow \text{ 5}\left(\text{a}\right)\\ Q\\ P\Leftarrow Q\end{array}$