Midterm 1 Flashcards
Natural Numbers
set of numbers {0,1,2,3…}
Integers
{-3,-2,-1,0,1,2,3}
Rational Numbers
Denoted by (Q) all numbers which may be written in the form p/q where p and q are integers
Real numbers
set of all rational and irrational numbers. and solution to x=sqrt(a) where a is an integer and xi is not rational
b|a
a=bx
prime numbers
p is prime provided p>1 and the only positive divisors of p are 1 and p
odd numbers
an integer a is odd provided there exists an integer x s.t a=2x+1
composite numbers
x is a member of the natural numbers >0. x is composite if there exists an integer b st. 1<b><x and b|x</b>
even definition
let x be an integer. x is even if 2|x
Conjectures
seem to be true but cannot be proven for all
converse of IF A THEN B
IF B THEN A
contrapositive of IF A THEN B
IF B’ THEN A’
inverse of IF A THEN B
IF ‘A THEN ‘B
Simple multiplication principle
if we have 2 element lists for which there are n choices for the 1st element and m choices for the 2nd. There are mn possible lists
extended multiplication pinciple
supp we have k element list where there are nj choices for the jth elemnt/ then there are n1n2n3..*nk possible lists
cardinality
the number of elements in a set
B proper subset A
there exists an x that is ina but not b
the power set
denoted 2^A is the set of all subsets of A
AuB=BuA AnB=BnA
communitive property
AuBuC=Au(BuC)
assosciative
Au(empty set)=
A
An(emptyset)=
empty set
Au(BnC)=(AuB)n(AuC)
distributive
inclusion exclusion principle
cardinality( AuB)= card(A) + card(B) - card(AnB)
set difference B-A
everything in B not in A
compliment of A
universe-A
Symmetric Difference (A triangle B)
(A-B)u(B-A)
Cartesian product AxB
{(a,b): a member of A, b member of B}
card(AxB)
card(B) x card(A)
Demorgans LAws
A-(BuC)= (A-B) N(A-C)
A-(BnC)=(A-B)u(A-C)
relations
a set of ordered pairs
x(R)y
means x is related to y
inverse relation of R={x,y: (x,y)}
R^(-1)={x,y: (y,x) member of R}
reflexive
if for all x in A (x,x) is in R
irreflexive
if for all x in A (x,x) is not in R
symetric
if (x,y) in R implies (y,x) in R
antisymetric
if (x,y) and (y,x) in R then x=y
transitive
if (x,y) and (x,z) are in R then (x,z) in R
