Algebra/Chapter 2/Logic and Proofs

Algebra

2.6: Logic and Proofs

---

Contructing a Proof of a Conditional Statement

Properties of Equality

Property Name	Addition	Subtraction	Multiplication	Division
Commutative	$a+b=b+a$	Doesn't work: $a-b\neq b-a$ This does: $a+(-b)=(-b)+a$	$a*b=b*a$	Doesn't work: $a/b\neq b/a$ This does: $a*1/b=1/b*a$
Associative	$(a+b)+c=a+(b+c)$	Doesn't work: $(a-b)-c\neq a-(b-c)$ This does: $(a-b)-c=a-(b+c)=a+(-b-c)$	$(a*b)*c=a*(b*c)$	Doesn't work: $(a/b)/c\neq a/(b/c)$ This does: $(a/b)/c=a*1/b*1/c=a/b*c$
Identity	$a+0=a$	$a-0=a$	$a*1=a$	$a/1=a$
Inverse	$a+-a=0$	$a-a=0$	$a*(1/a)=1$   as long as a ≠ 0.	$a/a=1$   as long as a ≠ 0.
Distributive	$a*(b+c)=a*b+a*c$	$a*(b-c)=a*b-a*c$	$a*(b+c)=a*b+a*c$	$(a+b)/c=a/c+b/c$ But wait: $a/(b+c)\neq a/b+a/c$

Practice Problems

Problem 2.80 (Using Properties of Numbers) Justify each step, using the properties of communativity and associativity in proving the following identities.

$a.\ (a+b)+(c+d)=(a+d)+(b+c)$
$b.\ (a+b)+(c+d)=(a+c)+(b+d)$
$c.\ (a-b)+(c-d)=(a+c)+(-b-c)$
$d.\ (a-b)+(c-d)=(a+d)-(b+c)$
$e.\ (a-b)+(c-d)=(a-d)+(c-b)$
$f.\ (a-b)+(c-d)=-(b+d)+(a+c)$
$g.\ ((a+b)+c)+d=(a+c)+(b+d)$
$h.\ (a-b)-(c-d)=(a-c)+(d-b)$

Problem 2.81 (Using Properties of Numbers) Determine if the following statements are true or false. Justify your conclusions.

a. If $a$, $b$, and $c$ are integers, then the number $ab+bc$ is an even number.
b. If $b$ and $c$ are odd integers, and $a$ is an integer, then the number $ab+bc$ is an even number.

Problem 2.82 (Using Properties of Numbers) We define an integer $a$ to be of

• Type I if $a=4n$ for some integer $n$
• Type II if $a=4n+1$ for some integer $n$
• Type III if $a=4n+2$ for some integer $n$
• Type IV if $a=4n+3$ for some integer $n$

a. Provide at least two examples of each of the four types of integers above.
b. Is it true that if $a$ is even, then it is of type I or III? Justify your answer.
c. Is it true that if $a*b$ is of type I, whenever $a$ or $b$ are of type III? Justify your answer.

Problem 2.83 (Using Properties of Numbers) For all real numbers $x$ and positive integers $n$, show that:

$(1-x)(1+x+x^{2}+...+x^{n-1}+x^{n})=1-x^{n+1}$