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)2 = x2 + 6x + 9 Ex. Sin2θ + cos2θ = 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

X2+3x-3x-6=0

X2=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)

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

However if a statement and its converse are true then we say that the hypothesis is equivalent to the

conclusion

HC and CH

This means H is true iff C is true

Ex. It can be shown that if a triangle has all 600 angles then it is equilateral.

Equilateral <==> all 600 angles

Note, when proving a theorem involving “iff”, you have to prove the statement both ways.

Ex. Prove that x2+y2= 0 iff x=0 and y=0

() given x2+y2=0, x,y≥0 x2=0, y2=0 x=0, y=0\

() given x=0, y=0 x2+y2=02+02=0

4) contrapositive

If we have a statement H then the contrapositive statement is: negation of C negation of H

Ex. If a triangle is equilateral then all its interior angles are 600

Contrapositive: if not all interior angles are 600 then a triangle is not equilateral.

A statement and its contrapositive are logically equivalent. Thus if we want we can prove the

contrapositive of a statement rather than proving the original statement.

H: 3n+7 is odd C: n is even

Contrapositive

Not C: n is odd Not H: 3n+7 is even

3n+7= 3(2k+1)+7

= 6k+3+7

=6k+10

=2(3k+5)

Since we have proved the contrapositive we have proved the original

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###### Document Summary

In mathematics a statement is a declarative sentence that can be true or false but not both. A mathematical identity is an equality relation that is always true for all values of the variable in the relation. Ex. (x+3)2 = x2 + 6x + 9 ex. An equation is an equality relation depending on some variables for which we are asked to find all values that make the statement true. Clearly not an identity since for x=0 -6=0. X= 6 you can always check by substituting. I think therefore i am if a triangle is equilateral then all interior angles are 60 . Prove that if m and n are even numbers then m+n is also even. The converse of a statement is a statement obtained by switching the hypothesis with the conclusion. If a statement is true it is not always guaranteed that its converse is true.

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