Commonly Used Mathematical
Notation
1 Logical Statements
Common symbols for logical statement:
_ logical disjunction: "or"
Note:
in mathematics this is always an "inclusive or"
i.e. "on or the other or both"
^ logical conjunction: "and"
: logical negation: "not"
! material implication: implies; if .. then
Note:
P !Q means:
if P is true then Q is also true;
if P is false then nothing is said about Q
can also be expressed as:
if P then Q
P implies Q
Q, if P
P only if Q
P is a su¢ cient condition for Q
Q is a necessary condition for P
sometimes writen as )
f :X!Y function arrow: function f maps the set X into the set Y
function composition: f g function such that (f g)(x) = f (g(x))
$ material equivalence: if and only if (i¤)
1
Note:
P $Q means:
means P is true if Q is true and P is false if Q is false
can also be expressed as:
P; if and only if Q
Q, if and only if P
P is a necessary and su¢ cient condition for Q
Q is a necessary and su¢ cient condition for P
sometimes writen as ,
is much less than
is much greater than
) therefore
8 universal quanti…cation: for all/any/each
9 existential quanti…cation: there exists
9! uniqueness quanti…cation: there exists exactly one
de…nition: is de…ned as
Note:
sometimes writen as :=
2
2 Set Notation
A set is some collection of objects. The objects contained in a set are known as
elements or members. This can be anything from numbers, people, other sets,
etc. Some examples of common set notation:
f; g set brackets: the set of ...
e.g. fa; b; cg means the set consisting of a, b, and c
fjg set builder notation: the set of ... such that ...
i.e. fxjP (x)g means the set of all x for which P (x) is true.
e.g. fn 2 N : n2 < 20g = f0; 1; 2; 3; 4g
Note: fjg and f:g are equivalent notation
; empty set
i.e. a set with no elements. fg is equivalent notation
2 set membership: is an element of
2
= is not an element of
2.1 Set Operations
Commonly used operations on sets:
[ Union
A[B set containing all elements of A and B.
A [ B = fx j x 2 A _ x 2 Bg
\ Intersect
A\B set containing all those elements that A and B have in common
3
A \ B = fx j x 2 A ^ x 2 Bg
n Di¤erence or Compliment
AnB set containing all those elements of A that are not in B
AnB = fx j x 2 A ^ x 2
= Bg
Subset
A B subset: every element of A is also element of B
A B proper subset: A B but A 6= B.
Superset
A B every element of B is also element of A.
A B A B but A 6= B.
2.2 Number Sets
Most commonly used sets of numbers:
P Prime Numbers
Set of all numbers only divisible by 1 and itself.
P = f1; 2; 3; 5; 7; 11; 13; 17:::g
N Natural Numbers
Set of all positive or sometimes all non-negative intigers
N = f1; 2; 3; :::g, or sometimes N = f0; 1; 2; 3; :::g
Z Intigers
Set of all integers whether positive, negative or zero.
Z = f:::; 2; 1; 0; 1; 2; :::g:
4
Q Rational Numbers
Set of all fractions
R Real Numbers
Set of all rational numbers and all irrational numbers p
(i.e. numbers which cannot be rewritten as fractions, such as , e, and 2).
Some variations:
R+ All positive real numbers
R All positive real numbers
R2 Two dimensional R space
Rn N dimensional R space
C Complex Numbers
Set of all number of the form:
a + bi
where:
a and b are real numbers, and
i is the imaginary unit, with the property i2 = 1
Note: P N Z Q R C
