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# Elements of logic this is the second days notes and covers contrapositives, hypothesis, converses, ect.

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Department
Mathematics
Course
MA100
Professor
Shane Bauman
Semester
Fall

Description
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 How
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