⇒
→
| material implication
| implies; if .. then
| propositional logic
|
A ⇒ B means: if A is true then B is also true; if A is false then nothing is said about B.
→ may mean the same as ⇒, or it may have the meaning for functions mentioned further down
|
| x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = 2 is in general false (since x could be −2)
|
⇔
↔
| material equivalence
| if and only if; iff
| propositional logic
|
| A ⇔ B means: A is true if B is true and A is false if B is false
|
| x + 5 = y + 2 ⇔ x + 3 = y
|
∧
| logical conjunction
| and
| propositional logic
|
| the statement A ∧ B is true if A and B are both true; else it is false
|
| n < 4 ∧ n > 2 ⇔ n = 3 when n is a natural number
|
∨
| logical disjunction
| or
| propositional logic
|
| the statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false
|
| n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number
|
¬
/
| logical negation
| not
| propositional logic
|
the statement ¬A is true if and only if A is false
a slash placed through another operator is the same as "¬" placed in front
|
| ¬(A ∧ B) ⇔ (¬A) ∨ (¬B); x ∉ S ⇔ ¬(x ∈ S)
|
∀
| universal quantification
| for all; for any; for each
| predicate logic
|
| ∀ x: P(x) means: P(x) is true for all x |
| ∀ n ∈ N: n2 ≥ n |
∃
| existential quantification
| there exists
| predicate logic
|
| ∃ x: P(x) means: there is at least one x such that P(x) is true
|
| ∃ n ∈ N: n + 5 = 2n
|
=
| equality
| equals
| everywhere
|
| x = y means: x and y are different names for precisely the same thing
|
| 1 + 2 = 6 − 3
|
:=
:⇔
| definition
| is defined as
| everywhere
|
x := y means: x is defined to be another name for y
P :⇔ Q means: P is defined to be logically equivalent to Q
|
| cosh x := (1/2)(exp x + exp (−x)); A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B)
|
{ , }
| set brackets
| the set of ...
| set theory
|
| {a,b,c} means: the set consisting of a, b, and c
|
| N = {0,1,2,...}
|
{ : }
{ | }
| set builder notation
| the set of ... such that ...
| set theory
|
| {x : P(x)} means: the set of all x for which P(x) is true. {x | P(x)} is the same as {x : P(x)}.
|
| {n ∈ N : n2 < 20} = {0,1,2,3,4}
|
∅
{}
| empty set
| empty set
| set theory
|
| {} means: the set with no elements; ∅ is the same thing
|
| {n ∈ N : 1 < n2 < 4} = {}
|
∈
∉
| set membership
| in; is in; is an element of; is a member of; belongs to
| set theory
|
| a ∈ S means: a is an element of the set S; a ∉ S means: a is not an element of S
|
| (1/2)−1 ∈ N; 2−1 ∉ N
|
⊆
⊂
| subset
| is a subset of
| set theory
|
A ⊆ B means: every element of A is also element of B
A ⊂ B means: A ⊆ B but A ≠ B
|
| A ∩ B ⊆ A; Q ⊂ R
|
∪
| set theoretic union
| the union of ... and ...; union
| set theory
|
| A ∪ B means: the set that contains all the elements from A and also all those from B, but no others
|
| A ⊆ B ⇔ A ∪ B = B
|
∩
| set theoretic intersection
| intersected with; intersect
| set theory
|
| A ∩ B means: the set that contains all those elements that A and B have in common
|
| {x ∈ R : x2 = 1} ∩ N = {1}
|
\\
| set theoretic complement
| minus; without
| set theory
|
| A \\ B means: the set that contains all those elements of A that are not in B
|
| {1,2,3,4} \\ {3,4,5,6} = {1,2}
|
( )
[ ]
{ }
| function application; grouping
| of
| set theory
|
for function application: f(x) means: the value of the function f at the element x
for grouping: perform the operations inside the parentheses first
|
| If f(x) := x2, then f(3) = 32 = 9; (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4
|
f:X→Y
| function arrow
| from ... to
| functions
|
| f: X → Y means: the function f maps the set X into the set Y
|
| Consider the function f: Z → N defined by f(x) = x2
|
N
| natural numbers
| N
| numbers
|
| N means: {0,1,2,3,...}
|
| {|a| : a ∈ Z} = N
|
Z
| integers
| Z
| numbers
|
| Z means: {...,−3,−2,−1,0,1,2,3,...}
|
| {a : |a| ∈ N} = Z
|
Q
| rational numbers
| Q
| numbers
|
| Q means: {p/q : p,q ∈ Z, q ≠ 0}
|
| 3.14 ∈ Q; π ∉ Q
|
R
| real numbers
| R
| numbers
|
| R means: {limn→∞ an : ∀ n ∈ N: an ∈ Q, the limit exists}
|
| π ∈ R; √(−1) ∉ R
|
C
| complex numbers
| C
| numbers
|
| C means: {a + bi : a,b ∈ R}
|
| i = √(−1) ∈ C
|
<
>
| comparison
| is less than, is greater than
| partial orders
|
| x < y means: x is less than y; x > y means: x is greater than y
|
| x < y ⇔ y > x
|
≤
≥
| comparison
| is less than or equal to, is greater than or equal to
| partial orders
|
| x ≤ y means: x is less than or equal to y; x ≥ y means: x is greater than or equal to y
|
| x ≥ 1 ⇒ x2 ≥ x
|
√
| square root
| the principal square root of; square root
| real numbers
|
| √x means: the positive number whose square is x
|
| √(x2) = |x|
|
∞
| infinity
| infinity
| numbers
|
| ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits
|
| limx→0 1/|x| = ∞
|
π
| pi
| pi
| Euclidean geometry
|
| π means: the ratio of a circle's circumference to its diameter
|
| A = πr² is the area of a circle with radius r
|
!
| factorial
| factorial
| combinatorics
|
| n! is the product 1×2×...×n
|
| 4! = 12
|
| |
| absolute value
| absolute value of
| numbers
|
| |x| means: the distance in the real line (or the complex plane) between x and zero
|
| |a + bi| = √(a2 + b2)
|
|| ||
| norm
| norm of; length of
| functional analysis
|
| ||x|| is the norm of the element x of a normed vector space
|
| ||x+y|| ≤ ||x|| + ||y||
|
∑
| summation
| sum over ... from ... to ... of
| arithmetic
|
| ∑k=1n ak means: a1 + a2 + ... + an
|
| ∑k=14 k2 = 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30
|
∏
| product
| product over ... from ... to ... of
| arithmetic
|
| ∏k=1n ak means: a1a2···an
|
| ∏k=14 (k + 2) = (1 + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 360
|
∫
| integration
| integral from ... to ... of ... with respect to
| calculus
|
| ∫ab f(x) dx means: the signed area between the x-axis and the graph of the function f between x = a and x = b
|
| ∫0b x2 dx = b3/3; ∫x2 dx = x3/3
|
f '
| derivative
| derivative of f; f prime
| calculus
|
| f '(x) is the derivative of the function f at the point x, i.e. the slope of the tangent there
|
| If f(x) = x2, then f '(x) = 2x
|
∇
| gradient
| del, nabla, gradient of
| calculus
|
| ∇f (x1, …, xn) is the vector of partial derivatives (df / dx1, …, df / dxn)
|
If f (x,y,z) = 3xy + z² then ∇f = (3y, 3x, 2z)
A transparent image for text is: Image:Del.gif ().
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