Proofs
A conjecture is a mathematical statement that has yet to be proven
A theorem is a mathematical statement that has been proven
Methods of proof:
- Proof by exhaustion
- Proof by deduction
- Proof by counterexample
- Proof by contradiction
- Consequence and equivalence
Consequence and equivalence
Notations
“If x = 3 then x2 = 9” is a conditional statement.
x is unknown but if x is 3 then x2 is 9
x = 3 implies x2 = 9
The symbol ⇒ means implies x = 3 ⇒ x2 = 9
The symbol ∴ means therefore x = 3 ∴ x2 = 9
This implication does not work in reverse because there are two possible values for x since 3 can be positive or
negative
Some implications can work in both directions
If x = 3 then x +1 = 4 can be written as :
x = 3 if and only if x + 1 = 4
If and only if is sometimes abbreviated to iff
A symbol for this is ⇔ which also means implies and is implied by
An equation is true for only a limited number of values of a variable
An identity is true for all values of a variable
3x = 12 is true only if x = 4
(x+1)(x-3) = 0 is only true for x = -1 and x = 3
(x+1)(x-3) ≡ x2 - 2x - 3 is true for all values of x
A two bar sign (=) is used for an equations and a three bar sign ( ≡ ) is used to emphasis that a statement is an
identity
EXAMPLES
x+2=4 ⇔ x=2
x=3 ⇒ x2 + 1 = 10
x2 is even ⇔ x is even
3
θ = 60o ⇒ sinθ = 2
A conjecture is a mathematical statement that has yet to be proven
A theorem is a mathematical statement that has been proven
Methods of proof:
- Proof by exhaustion
- Proof by deduction
- Proof by counterexample
- Proof by contradiction
- Consequence and equivalence
Consequence and equivalence
Notations
“If x = 3 then x2 = 9” is a conditional statement.
x is unknown but if x is 3 then x2 is 9
x = 3 implies x2 = 9
The symbol ⇒ means implies x = 3 ⇒ x2 = 9
The symbol ∴ means therefore x = 3 ∴ x2 = 9
This implication does not work in reverse because there are two possible values for x since 3 can be positive or
negative
Some implications can work in both directions
If x = 3 then x +1 = 4 can be written as :
x = 3 if and only if x + 1 = 4
If and only if is sometimes abbreviated to iff
A symbol for this is ⇔ which also means implies and is implied by
An equation is true for only a limited number of values of a variable
An identity is true for all values of a variable
3x = 12 is true only if x = 4
(x+1)(x-3) = 0 is only true for x = -1 and x = 3
(x+1)(x-3) ≡ x2 - 2x - 3 is true for all values of x
A two bar sign (=) is used for an equations and a three bar sign ( ≡ ) is used to emphasis that a statement is an
identity
EXAMPLES
x+2=4 ⇔ x=2
x=3 ⇒ x2 + 1 = 10
x2 is even ⇔ x is even
3
θ = 60o ⇒ sinθ = 2
