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Math 381
Exam 1
22
Mathematics
Undergraduate 3
02/10/2009

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Term
Statement
Definition
p--> q
Logically equivalent to contrapositive
Term
converse
Definition
q-->p
Logically equivalent to inverse
Term
Inverse
Definition
¬p-->¬q Logically equivalent to converse
Term
Contrapositive
Definition
¬q--> ¬p logically equivalent to statement
Term
negation
Definition
p^¬q
Term
Proof by contrapositive
Definition
(a-->b)=(¬b-->¬a)
Assume ¬b, then demonstrate this assumption forces ¬a.
Term
proof by contradiction
Definition
we show that ¬(a-->b) is impossible. Assume 'not b', then show a contradiction if a^¬b were to be the statement.
Term
direct proof
Definition
Assume 'a', show that it forces 'b'.
Term
non-constructive existence proof
Definition
we prove some specific object exists by showing it can't not exist
Term
Proof by mathematical induction
Definition
to prove statement P(n)
1. verify p(0) is true
2. assume p(k) is true for some generically specified k
3. show p(k) true--> p(k+1) true
Term

p^(q V r)

 

Definition
(p^q) V (p^r)
Term
p-->q
Definition

¬q-->¬p AND

¬p V q

Term
p V q
Definition
¬p --> q
Term
p^q
Definition
¬(p-->¬q)
Term
¬(p-->q)
Definition
p^¬q
Term
pV¬p
Definition
True
Term
p^¬p
Definition
False
Term
Reflexive
Definition
A relation R on a set A is called reflexive if (a,a)E R for every element aEA.
Term
Symmetric
Definition
A relation R on a set A is symmetric if (b,a) E R whenever (a,b) E R, for all a,b E A.
Term
antisymmetric
Definition
A relation R on a set A such that for all a,b E A if (a,b) E R and (b,a)ER, then a=b is called antisymmetric
Term
transitive
Definition

A relation R on a set A is called transitive if whenever (a,b) E R and (b,c)E R, then (a,c) E R, for all a, b, c E A.

 

The relation R on a set A is transitive if and only if Rn C R for n=1, 2, 3,..

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