Class 11th mathematics (relations and functions)

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If P and Q are non-empty sets then the set of all ordered pairs (a, b) is called the Cartesian product of A and B [where a ∈ P and b ∈ Q]. It can be represented symbolically as P × Q = {(a, b) | a ∈ P and b ∈ Q}.
Example
If P = {3, 4, 5} and Q = {6, 7}, then
  • P × Q = {(3, 6), (4, 6), (5, 6), (3, 7), (4, 7), (5, 7)}
  • Q × P = {(6, 3), (6, 4), (6, 5), (7, 3), (7, 4), (7, 5)}
Case 1:  Two ordered pairs are said to be equal if their corresponding first and second elements are equal, i.e. (p, q) = (m, n) if p = m and q = n.
Case 2: If n(P) = a and n (Q) = b, then n(P × Q) = a × b.
Case 3: If P × P × P = {(p, q, r) : p, q, r ∈ P}. Then, (p, q, r) is known as an ordered triplet.

When Sets are said to be in a Relation?

If P and Q are two non-empty sets, then a Relation (R) from set P to set Q is a subset of set P × Q. In this relation, the set of all first elements in R is known as the domain of the relation (R) and the set of all second elements is known as the range of R.
  • A relation (R) can be represented in either Roster or set builder form. The visual representation of a relation is done using an arrow diagram.
  • If n(P) = a, n(Q) = b; then n(P × Q) = ab. Also, the total possible relations from set P to Q = 2ab.
For example: The set R = {(11, 12), (-12, 13), (11/2, 13)} is a relation. The domain = {11, -12,11/2} and its range = {12, 13}

What are the Functions?

A relation from set P to Q is said to be a function if all the elements of set P have just one image in set Q. The expression f : P → Q denotes: f is a function from P to Q and the Domain and co-domain of function(f) are represented by P and Q respectively.

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