Discrete Math Definitions

steps of a direct proof P=>Q

1. assume P is true 2. Use P to that Q is true

If a and b are consecutive numbers then the sum of a and b is odd. use a contradiction

Assume a+b is not odd. Let k be an element of integers. If it's not odd then no integer k such that the a+b=2k+1 but a+b=a+a+1=2a+1. Shown that a+b !=2k+1 but a+b=2a+1. By default a+b is odd.

If a and b are consecutive numbers then the sum of a and b is odd. use a direct proof

Assume a,b are consecutive and b=a+1. Let k be an integer. So a+b= a+a+1 and a+b=2a+1.

If a and b are consecutive numbers then the sum of a and b is odd. use contraposition

Assume that sum a+b is not odd. There does not exist and integer k, a+b=2k+1. So a+b=k+k+1 does not hold for any integer k, but since k+1 is the successor of k, this implies that a and b cannot be consecutive.

4 basic proofs techniques used in mathematics

Direct Proof, Proof by contradiction, Proof by induction, proof by contraposition

Two integers a and b are consecutive if and only if

b=a+1

Proof by contradiction means that P=>Q is

both false and true

An integer number n is even if and only if there exists an integer k such that

n=2k

An integer number n is odd if and only if there exists an integer k such that

n=2k+1

Contrapositive means that if P=>Q then

not Q => not P

Proof means that a definition is

satisfied

Prepositions in math assume that a statement is either (prepositions are well defined)

true or false