Discrete Mathematics

Properties/Theorems

Property 2.1

Closure: performing an operation on a set yields numbers also in the set

Communicative laws: order of addition and multiplication can be swapped

Associate laws: parentheses can be moved

Distributive law: can distribute

Additive and Multiplicative Identities: Can add by 0 or multiply by 1

Additive Inverse: there is a number that adds to zero for a+(-a)=0

Multiplicative Inverse: there is a number a*a^-1 that equals 1

Exponential laws: exponet rules, (aⁿ)^m=a^nm and (ab)ⁿ=aⁿbⁿ and aⁿa^m

Theorem 2.2

Additive inverses are unique

0 is the unique additive identity for all real numbers

The multiplicative inverse of a is unique if a is not 0

1 is the unique multiplicative identity for all real numbers

A double negative of a number is the positive version of the number

A negative can be distributed

If a is not 0, then if ab=ac, b=c

Zero product property: If ab=0, then a=0 or b=0

Even/Odd

Even: n=2j

Odd: m=2k+1

Divides

If there exists an integer q such that a=bq (and b is not 0)

Division Theorem (aka Algorithm)

For any two integers a and b, when b is not 0, there are integers p and r such that 0<=r<|b| and a=qb+r