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In mathematics, a relation is a set of ordered pairs (a, b), where a and b are elements of a set. The set of all first elements (a) of the ordered pairs is called the domain of the relation, and the set of all second elements (b) is called the range. A function is a specific type of relation in which each element of the domain corresponds to exactly one element in the range.A function is often represented by an equation, such as y = f(x), where x is the input (or domain) and y is the output (or range). The function f(x) assigns a unique output value, y, for each input value, x.For example, the relation {(1,2), (3,4), (5,6)} is not a function because the first element 1 is repeated, while the function f(x)=x+1 is a function because for each input value x, there is exactly one output value f(x)=x+1.

a relation on a set is a reflexive if every element in the set is related to itself. A relation is symmetric if for every ordered pair (a,b) in the relation, the ordered pair (b,a) is also in the relation. A relation is transitive if for every ordered pair (a,b) and (b,c) in the relation, the ordered pair (a,c) is also in the relation.A reflexive relation is denoted by (a,a) ∈ R for all a ∈ A, where R is the relation and A is the set.A symmetric relation is denoted by (a,b) ∈ R implies (b,a) ∈ RA transitive relation is denoted by (a,b) ∈ R and (b,c) ∈ R implies (a,c) ∈ RFor example, the relation "is equal to" is reflexive because every number is equal to itself, symmetric because if a = b then b = a, and transitive because if a = b and b = c then a = c.A relation can be reflexive and symmetric but not transitive and also can be reflexive and transitive but not symmetric.

An equivalence relation is a relation that is reflexive, symmetric, and transitive. In other words, it is a relation that defines a partitioning of a set into equivalence classes.An example of an equivalence relation is "is equal to" on the set of real numbers. This relation is reflexive because every number is equal to itself, symmetric because if a = b then b = a, and transitive because if a = b and b = c then a = c.Another example of an equivalence relation is "has the same parity" on the set of natural numbers. This relation is reflexive because every number is of the same parity as itself, symmetric because if a and b have the same parity, then b and a do as well, and transitive because if a and b have the same parity and b and c have the same parity then a and c have the same parity.In general, given a set A, an equivalence relation on A is a relation R on A such that 1. it is reflexive, i.e., (a,a) ∈ R for all a ∈ A 2. it is symmetric, i.e., (a,b) ∈ R implies (b,a) ∈ R 3. it is transitive, i.e., (a,b) ∈ R and (b,c) ∈ R implies (a,c) ∈ REquivalence relations have many important uses in mathematics, including in number theory, group theory, and topology.

A function is said to be one-to-one (or injective) if it assigns a unique output value for each input value in its domain. In other words, for any two distinct input values, the function assigns two distinct output values.A function f from A to B is said to be injective if for all a1, a2 in A, if f(a1) = f(a2) then a1 = a2.For example, the function f(x) = x^2 is one-to-one because for any two distinct input values x1 and x2, the output values will also be distinct (e.g. f(2) = 4 and f(3) = 9).On the other hand, the function f(x) = x^3 is not one-to-one because there are distinct input values that map to the same output value (e.g. f(-2) = -8 and f(2) = 8).Injective functions are also known as "injections" or "one-to-one functions" and are often represented with a symbol of a arrow with a small vertical bar above the arrowhead.A function which is not one-to-one is called "many-to-one" function.

A function is said to be onto (or surjective) if for every element in the range of the function, there is at least one element in the domain that maps to it. In other words, every element of the codomain is "hit" or "covered" by the function.A function f from A to B is said to be surjective if for all b in B, there exists an a in A such that f(a) = b.For example, the function f(x) = 2x is onto (or surjective) because for any integer y, there exists an integer x such that f(x) = 2x = y.On the other hand, the function f(x) = x^2 is not onto (or surjective) because there are elements in the range (non-negative real numbers) that are not mapped from any element in the domain (real numbers)A function which is not surjective is called "not-onto" function.A function that is both one-to-one (injective) and onto (surjective) is called a bijective function.

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