Relations and Functions
A pair of elements grouped together in a particular order is called an ordered pair.
A relation R from a non-empty set X to a non-empty set Y is a subset of the cartesian product X × Y.
The set of all first elements of the ordered pairs in a relation R from a set X to a set Y is called the domain of the relation R.
The set of all second elements in a relation R from a set X to a set Y is called the range of the relation R. The whole set Y is called the codomain of the relation R.
A relation is represented algebraically either by the Roster method or by the Set-builder method.
A relation can be visually represented by an arrow diagram.
A relation R from B to B is also stated as a relation on B.
A relation f from a set X to a set Y is said to be a function if every element of set X has one and only one image in set Y.
A function f is a relation from a non-empty set X to a non-empty set Y such that the domain of f is X and no two distinct ordered pairs in f have the same first element.
A function which has either R or one of its subsets as its range is called a real valued function. If its domain is also either R or a subset of R, it is called a real function.
Key words: Cartesian Products of Sets, Identity function, Constant function, Polynomial function, Rational functions, Modulus function, Signum function, Greatest integer function, Algebra of real functions, Addition of two real functions, Subtraction of a real function from another, Multiplication by a scalar, Multiplication of two real functions, Quotient of two real functions, Pointwise multiplication