Peano's Axiom 1 (for a set N)
N has a distinguished element which we call '1'
Peano's Axiom 2 (for a set N)
There exists a distinguished set map o- : N -> N
Peano's Axiom 3 (for a set N)
o- is one to one
Peano's Axiom 4 (for a set N)
There does not exist an element n E N s.t. o-(n) = 1
Peano's Axiom 5 (for a set N)
(Principle of Induction) If S ( N s.t. a) 1 E S and b) for every n E S, o-(n) E S, then S = N
addition on N
1. For all n E N, n + 1 = o-(n); 2. For any n, m E N, n + o-(m) = o-(n+m)
associativity on N
If x, y, z E N, then x + (y + z) = (x + y) + z
commutativity of addition on N
For any x, y E N, x + y = y + x
cancellative law on N
for any x, y, z E N, if x + z = y + z, then x = y
distributive law on N
for any x, y, z E N, then x (y + z) = x y + x z and (y + z) x = y x + z x
associative law for multiplication
For any x, y, z E N, x(yz) = (xy)z
property of 1 (rule 4)
For any x E N, 1 * x = x
commutative law for multiplication
For any x,y E N, xy = yx
n less than m (ordering on N) (n < m)
If n, m E N, we say that _________________ if there exists a k E N s.t. m = n + k
n less than or equal to m (ordering on N)
either n = m or n < m
least element in N
Let S ( N. Then an element n E S s.t. for all m E S, n </= m
cancellative law for multiplicaiton
for any x, y, z E N, if x z = yz, then x = y
uniqueness of identity
for some x, y E N, xy = x, then y =1
relation R on a set S
a subset of S x S; R ( S x S
equivalence relation R on set S
a relation that satisfies the following conditions: 1. For all a ES, a ~ a (Reflexivity); 2. If a ~ b, then b ~ a (Symmetry); 3. If a ~b and b ~ c, then a ~ c (Transitivity)
equivalence relation associated to the function f
If f: A -> B is a function, then the relation s.t. a ~ a' if f (a) = f(a')
addition on Z
If A, B E Z, then (since A, B are subsets of S), we pick elements (a, b) E A, and (c,d) E B and A + B = [(a + c, b + d)]
multiplication on Z
A, B E Z = [(ac + bd, ad + bc)]
associativity, commutativity, and cancellative properties of addition
same as N
zero
For any two natural numbers a, b E N, (a, a) ~ (b, b) and [(a, a)] = [(b, b)], denoted by 0; For any A E Z, A + 0 = A = 0 + A
additive inverse in Z
If A = [(a,b)], we denote -A = [(b, a)]; A + (-A) = 0; unique
distributivity, associative, and commutative laws of multiplication in Z
same as N
one
For a, b E N, (o-(a), a) ~ (o-(b), b); the equivalence class [(o-(a), a)] for any a E N; For all A E Z, A x 1 = A = 1 x A
cancellative law for multiplication
If A, B, c E Z and A x C = B x C with B nonzero, then A = B
positive integer
If A E Z it is _____ if 0 < A
nonnegative integer
If A E Z, it is ____ if 0 </= A.
negative integer
If A E Z, it is ______ if A < 0
absolute value of n E Z
IaI = a if a is non-negative and IaI = -a if a is negative
subtraction on Z
If a, b E Z then a - b = a + (-b)
addition on Q
If a,c E Z and b, d E Z- {0}, then [(a, b)] + [(c,d)] = [ad + bc, bd],
multiplication on Q
If a,c E Z and b, d E Z- {0}, then [(a, b)] x [(c,d)] = [ac, bd]
properties of multiplication, addition, and subtraction on Q
Same as on Z
division in Q
For a, b E Q and b nonzero, then If A = [(a,b)] and B = [(c,d)], then A/B = [(ad, bc)]; c also nonzero
sequence of rational numbers
a set map f: N -> Q; we are given rational numbers xn for every natural number n {xn}
Cauchy Sequence
a sequence where, if given any 0< e E Q, there exists a natural number N (which will depend on e) s.t. for all n, m >/= N, natural numbers, we have Ixn - xmI < e
real numbers
the set of equivalence classes of Q
addition in R
If A = [{xn} and B = [{yn}] then A + B = [{xn + yn}]
multiplication in R
If A = [{xn} and B = [{yn}] then A x B = [{xn *yn}]
A less than B in R (ordering on R)
If {xn} and {yn} are CS and A = [{xn} and B = [{yn}] then A < B if there exists an e > 0, rational number and an N E N s.t. for all n, m >/= N, ym - xn > e
A less than or equal to B in R (ordering on R)
A < B or A = B
lower bound a for S
If S ( R and a E R then a is ______ if a </= s for all s E S.
upper bound b for S
If S ( R and b E R then b is a ________ if s </=b for all s E S
infimum of S
If S ( R is a nonempty subset and has a lower bound M then the real number & (called the _________ of S) s.t. for any s E S, s >/= & ; greatest lower bound for S;
supremum of S
For any s E S, s </= { ; least upper bound for S
Cauchy Sequence in R
A sequence of real numbers {An} is a CS if for any positive real number e there exists an N E N s.t. for all n, m >/= N, IAn - AmI < e
Cauchy Sequence in R (II)
A sequence of real numbers {An} is a ________ if for any positive rational number e there exists an N E N s.t. for all n, m >/= N, IAn - AmI < e
Limit of the sequence {xn}
If {xn} is a CS of real numbers and x is the unique real number from theorem 4.3, then x is the _______ written x = limxn
continuous at a point x E R
A function f: R -> R is called ________ if for any CS, with lim xn = x, the sequence {f(xn)} is a CS