Question

Define a function as corresponding to two sets.

Answer 1

We have to know the set theory and the definition of Relation of sets to define a function. A function is a special relation between two sets.

Complete step by step solution:
A function is a special relation between the elements of two sets such that every element of the first set will have a unique image in the second set. So, every element of set A will have one and only one image in set B.
For example, Let A and B be two sets containing elements \left\\{ {1,2,3} \right\\}and \left\\{ {1,2,3,4,5,6,7,8,9,10} \right\\}respectively. Let us consider a relation $f$from set A to B , defined as $f:A \to B$ such that every element a of A, has one and only one image $f(a)$in B defined as $f(a) = 3a\forall a \in A$. Then the elements of the function will be:
f(1) = 1 \times 3 = 3 \\\ f(2) = 2 \times 3 = 6 \\\ f\left( 3 \right) = 3 \times 3 = 9 \\\
Which belong to B.
That is, every element of A has one and only one image in B. Hence, $f$ is a function
However not every image in B has a pre- image in A.
The set A is called the Domain of the function $f$ while the set B becomes the Range of the function $f$.

Note: Function is not to be confused with Relation. Apart from equal functions there are other types of functions also. Let us consider two sets A and B. A one-to-one function is one where every element belonging to a set A, that has exactly one unique image in set B. An onto function is one where every element belonging to set B has at least one preimage in the set A. A function which is both one-to-one and onto, is called an injective function. If either of the characteristics is missing, the function will not be an injective function.