A (binary) relation R:X→Y is a sub-set of the Cartesian product X × Y.
ie, R ⊆ (X × Y)
A set of ordered pairs.
n-ary relation: a sub-set of a Cartesian product A₁ × A₂ × … × Aₙ
A relation R that is:
Eg: Arithmetic equality.
Given an equivalence relation R defined on a set A, if a ∈ A, equivalence class of a is
[a] = {x∈A | aRx}
Set of all equivalence classes with regard to an equivalence relation R defined on a set A is
[A]= {[a] | a ∈ A}
which is also written as A/R (read as A modulo R). This is the quotient set of A by R.
Not the same as relation.
A binary relation is not necessarily a function
R ⊆ (X × X)
but a binary operation is a function like
R : (X × X) → X
A relation that is
A relation that is
ie, a partial order is an anti-symmetric preorder.
Combination of a set and a partially ordered relation on that set.
Eg: the relation <= on the set of integers.
a<=a ie, reflexive.
a<=b and b<=a means a==b. ie, anti-symmetric
a<=b and b<=c means a<=c. ie, transitive
A poset where for any two elements a and b of the set, aRb or bRa.
Eg: <= over the set of integers is a total ordering. But < over integers isn't. Because if a=b, neither a<b nor b<a holds.
A special type of relations.
A relation R:X→Y is a function iff every elemment of X has only one image in Y.
Every element in X must have exactly one image in Y.
Also denoted as
y = f(x)
Functions are useful to represent how one quantity changes with respect to changes in another quantity.
Like the value of a sinsuoidal wave (of the form A.sin(ωt + Φ)) being dependent on time (t).
A relation is not a function if one of the following is true:
If for y = f(x),
The domain of a relation R:X→Y is the set X.
See: https://math.stackexchange.com/questions/3317941/what-is-the-difference-between-codomain-and-range
The range of a relation R:X→Y is
range(R) = {R(x) | x∈X}
—
Codomain of a relation R is a set that contains range(R).
Range ⊆ Codomain
Codomain is kind of 'more independent'.
Eg: In a function f:X→Y where X = {1, 2}, Y = {10, 20, 30} and f:X→Y = {1↦10, 2↦20},
The inverse of a function f:X→Y is f⁻¹:Y→X where (x, y) ∈ f mean (y, x) ∈ f⁻¹.
ie, g is inverse of f if f composed with g gives the identity function.
There may be left inverse and right inverse (function here, not element).
g∘f = id (g is left inverse of f)
f∘g = id (g is right inverse of f)
When the domain of a function g and the codomain of another function f are the same, we can compose f and g.
f:A->B
g:B->C
then
g∘f = A -> C
(Notice that the order in which function are written look as if it is in 'reverse order').
Each element in the codomain has only one pre-image in the domain.
Whenever f(a1) = f(a2), a1 = a2.
If a is the set of husbands and b is the set of wives, if R is an injective function, then there is no polyandry. 🥱
Every element in the codomain has at least one pre-image in the domain.
∀b∈B, ∃a∈A, f(a) = b
If a is the set of men and b is the set of women, if R is a surjective function (indicative of marriage), then every woman is married.
A function that is both one-to-one and onto.
An bijective function is invertible.
A function can have an inverse iff it is a bijection (since it sort of needs to go somewhere and come back).
If f:A→B, then f⁻¹:B→A.
A partial function is a function that is defined only on part of its domain. ³
This is what we usually mean when we say 'function'.
f: ℝ → ℝ such that ∀x, f(x) = x.
Its graph is a straight line passing through the origin at 45°.
f: ℝ → ℝ such that ∀x, f(x) = c where c is a constant.
Its graph is a straight line parallel to the X axis, intercepting the Y axis at y=c.
f: ℝ → ℝ such that f(x) = a₀ + xa₁ + x²a₂ + .... + xⁿaₙ where n is a non-negative integer and all aᵢ ∈ ℝ.
Examples:
Whereas 2x¾ + 4x⁴ is not a polynomial function since the power x is not a whole number.
A restricted form of polynomial function.
Has the form f(x) = ax² + bx + c where
Degree of term with highest degree is 2.
A bivariate version (ie, 2 variables) would be f(x,y) = ax² + by² + cxy + dx + ey + f.
Functions of the form f(x)/g(x) where
Coefficients of the polynomial terms needn't be rational??
Eg: (x³ - x²) / x³
A function f:ℝ→ℝ
f(x) = { x, if x ≥ 0
-x, otherwise}
Graph is V-shaped, with point of the base of the 'V' being at the origin and the arms of the 'V' being at 45° with the origin.
|
|
|
\ | /
\ | /
\ | /
\ | /
\ | /
\|/
---------+---------
O
A function f:ℝ→ℝ
f(x) = { 1, if x>0,
0, if x==0,
-1, if x<0
Graph looks like a step.
┃
┃
1┃------------
┃
┃
━━━━━━━━━━━━⊙━━━━━━━━━━━━━
0┃
┃
----------┃ -1
┃
┃
Though range is ℝ, co-domain = {1, 0, -1}
1 eˣ
f(x) = --------- = --------
1 + e⁻ˣ eˣ + 1
Graph is a S-shaped curve, with one end 'tending' to 0 and the other 'tending' to 1.
Sigmoid function has an interesting property:
f(x) = 1 - f(-x)
A function f:ℝ→ℝ (or f:ℝ→ℤ maybe?)
Denoted by [x].
f(x) = [x]
Value is the biggest integer ≥ x.
Examples:
lim f(x)
x→k
is the limit of the function f at a point k, which lies in the domain of f.
f is said to have a limit at k if the left and right limits of f at k exists and are the same:
lim f(x) = lim f(x) = lim f(x) = L
x→k x→k⁻ x→k⁺