Algebra/Chapter 2/Logic and Proofs

Algebra

2.5: Logic and Proofs

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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: $a/(b+c)\neq a/b+a/c$