Relations and Functions in Math

Relation and Function

• A relation R on a set A is symmetric if (a, b) ∈ R implies (b, a) ∈ R.
• A relation R on a set A is reflexive if (a, a) ∈ R for each a ∈ A.
• A relation R on a set A is transitive if (a, b) ∈ R and (b, c) ∈ R imply (a, c) ∈ R.

Examples of Relations

• Let A = {1, 2, 3, 4, 5, 6, 7}. The relation R = {(x, y) ∈ A × A : x + y = 7} is symmetric but neither reflexive nor transitive.
• Let A = {1, 2, 3, 4, 5, 6, 7}. The relation R = {(x, y) ∈ A × A : x + y = 7} is not an equivalence relation.

Functions

• A function f : A → B is bijective if it is both injective and surjective.
• A function f : A → B is injective if for every b ∈ B, there exists at most one a ∈ A such that f(a) = b.
• A function f : A → B is surjective if for every b ∈ B, there exists at least one a ∈ A such that f(a) = b.

Examples of Functions

• Let f : N → Z be defined by f(x) = x^2 + 3x - 2. Then f(x) is not injective but surjective.
• Let f : [2, 3] → R be defined by f(x) = x^3 + 3x - 2. Then the range of f(x) is contained in the interval [1, 12].
• Let f : R → R be defined by f(x) = x^2 - 6x - 14. Then f^(-1)(2) equals {-2, 8}.

Inverse of Functions

• The inverse of a function f is denoted by f^(-1).
• The inverse of a function f is a function g such that g(f(x)) = x for every x in the domain of f.

Periodic Functions

• A function f is periodic if there exists a positive real number T such that f(x + T) = f(x) for every x in the domain of f.
• The period of a function f is the smallest positive real number T such that f(x + T) = f(x) for every x in the domain of f.

Examples of Periodic Functions

• The function f(x) = sin(x) is periodic with period 2π.
• The function f(x) = cos(x) is periodic with period 2π.
• The function f(x) = tan(x) is periodic with period π.

Graphs of Functions

• The graph of a function f is the set of all points (x, y) such that y = f(x).
• The graph of a function f can be used to determine the domain and range of f.

Examples of Graphs of Functions

• The graph of the function f(x) = x^2 is a parabola.
• The graph of the function f(x) = sin(x) is a sine wave.
• The graph of the function f(x) = cos(x) is a cosine wave.

Equivalence Relations

• An equivalence relation on a set A is a relation R that is reflexive, symmetric, and transitive.
• An equivalence relation on a set A partitions A into equivalence classes.

Examples of Equivalence Relations

• The relation R = {(x, y) ∈ A × A : x - y is an integer} is an equivalence relation on the set of real numbers.
• The relation R = {(x, y) ∈ A × A : x^2 = y^2} is an equivalence relation on the set of real numbers.

Reflexive, Symmetric, and Transitive Relations

• A relation R is reflexive if (a, a) ∈ R for every a ∈ A.
• A relation R is symmetric if (a, b) ∈ R implies (b, a) ∈ R.
• A relation R is transitive if (a, b) ∈ R and (b, c) ∈ R imply (a, c) ∈ R.

Examples of Reflexive, Symmetric, and Transitive Relations

• The relation R = {(x, y) ∈ A × A : x + y = 7} is symmetric but neither reflexive nor transitive.
• The relation R = {(x, y) ∈ A × A : x - y is an integer} is reflexive, symmetric, and transitive.

Identify properties of relations such as symmetry, reflexivity, and transitivity in set theory. Apply these concepts to examples of relations on sets.