Algebra/Chapter 2/Logic and Proofs

 Algebra/Chapter 2 ← Real Numbers Logic and Proofs

2.6: Logic and Proofs

Contructing a Proof of a Conditional Statement

Properties of Equality

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

Practice Problems

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

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

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

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

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

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

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

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

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