Notation Page
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Analysis Name: June 14, 2015 MATH 317 “What did that mean again?”: Notation 1 Number Sets • N: The natural numbers {1, 2, 3,... }, excluding 0. • Z: The integers {..., -3, -2, -1, 0, 1, 2, 3,... }. • Q: The rational number A.K.A. {p/q | p, q ∈ Z and q 6=0}. • R: The real numbers A.K.A Rational numbers and irrational numbers. • φ or ∅: The empty set, or the set containing no elements; not the number zero 0. 2 Set Builder Notation • x ∈ S : x is an element of the set S . • x/ ∈ S : x is not an element of the set S • A ⊆ B: Set A is a subset of set B • A = B: Set A is equal to set B... i.e. A ⊆ B and A ⊇ B. • A ⊂ B or A ( B : set A is a proper subset of set B ... i.e. A ⊆ B and A 6= B. • A ∩ B: The intersection of set A and B ... i.e. elements that are in both A and B. • A ∪ B: The union of set A and B ... i.e. elements are in either A or B. • ∞ \ i=1 A i or \ i∈N A i : elements that are in A i for all i ∈ N. • ∞ [ i=1 A i or [ i∈N A i : elements that are in some A i for i ∈ N. • {-1, 2, 5}: the set containing the elements -1, 2, and 5. •{ ‘elements’ | ‘properties’ } or { ‘elements’ : ‘properties’ }: the set of ‘elements’ such that ‘properties’ hold.
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Transcript of Notation Page
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Analysis Name:June 14, 2015 MATH 317
What did that mean again?: Notation
1 Number Sets
N: The natural numbers {1, 2, 3, . . . }, excluding 0. Z: The integers {. . . ,3,2,1, 0, 1, 2, 3, . . . }. Q: The rational number A.K.A. {p/q | p, q Z and q 6= 0}. R: The real numbers A.K.A Rational numbers and irrational numbers. or : The empty set, or the set containing no elements; not the number zero 0.
2 Set Builder Notation
x S: x is an element of the set S. x / S: x is not an element of the set S A B: Set A is a subset of set B A = B: Set A is equal to set B . . . i.e. A B and A B. A B or A ( B : set A is a proper subset of set B . . . i.e. A B and A 6= B. A B: The intersection of set A and B . . . i.e. elements that are in both A and B. A B: The union of set A and B . . . i.e. elements are in either A or B.
i=1
Ai oriN
Ai: elements that are in Ai for all i N.
i=1
Ai oriN
Ai: elements that are in some Ai for i N.
{1, 2, 5}: the set containing the elements 1, 2, and 5. { elements | properties } or { elements : properties }: the set of elements such
that properties hold.
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Ac: The set of elements not in set A. A \B: elements in A that are not in B supA: The supremum of the set A. inf A: the infimum of the set A.
3 Function Notation
f : A B: The function f with domain A and codomain B. f(a): For a function f : A B, f(a) is the unique element in B that is the image of
the element a A through f . f : A B: The injective (or one-to-one) function f with domain A and codomain B f : A B: The surjective (or onto) function f with domain A and codomain B. f1(S): The preimage of the set S.
4 Algebra, Arithmetic, etc.
p|q: p divides q, i.e. q = pr for some r N. (an)nN: the sequence of elements an for n N.
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