Elements of Logic
In mathematics a statement is a declarative sentence that can be true or false but not both.
Ex. 7 is a prime number – true
Ex. (-3, 4] εŽ – false
A mathematical identity is an equality relation that is always true for all values of the variable in the
relation. Ex. (x+3) = x + 6x + 9 Ex. Sin θ + cos θ = 1
An equation is an equality relation depending on some variables for which we are asked to find all
values that make the statement true.
Ex. Solve x(x+3) – 3(x+2) = 0
Clearly not an identity since for x=0 -6=0
2
X +3x-3x-6=0
X =6
X=±√6 – you can always check by substituting.
Solution: {-√6, √6}
(read (2) what is mathematics? Proof and solutions)
Proofs: 1) identities LS=RS
Ex. Prove ( )( )
( )
( )( ) 2) hypothesis/conclusion
Ex. “I think therefore I am” “if a triangle is equilateral then all interior angles are 60⁰
Ex. Prove that if m and n are even numbers then m+n is also even
Proof: givens: m, n are even
m=2s, n=2t s,tεŽ
m+n= 2s+2t= 2(s+t)
e e
The converse of a statement is a statement obtained by switching the hypothesis with the conclusion
Ex. I think therefore I am
Converse: I am therefore I think
If a statement is true it is not always guaranteed that its converse is true.
Ex. If m+n is even it is not guaranteed that m, n are even. Counter example: 8=3+5
3) Equivalent statements
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