| 87 - 36 = 51
⇒
→
| 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||
| ∑
| addition
| 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
| ∏
| multiplication
| 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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