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 [were a ∈ P and b ∈ Q]. It can be represented symbolically as P × Q = {(a, b) | a ∈ P and b ∈ Q}.
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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 of 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 codomain of function (f) are represented by P and Q respectively.
Relations and Functions Class 11 Practice Questions
- Let P = {-11, 12, 13} and Q = {11, 23}. Determine
- P × Q
- Q × P
- Q × Q
- P × P
- If A = {y : y < 4, y ∈ N}, B = {y : y ≤ 2, y ∈ W(set of whole numbers)}. Find (A ∪ B) × (A ∩ B).
- If P = {y : y ∈ W, y < 4}, Q = {y : y ∈ N, 2 < y < 6}, and R = {3, 6}. Find (i) P × (Q ∩ R) (ii) P × (Q ∪ R).
- Find the values of p and q, if (2p + q, p – q) = (8, 3).
- Given P = {11, 12, 13, 14, 15}, S = {(a, b) : a ∈ P, b ∈ P}. Find the ordered pairs satisfying:
- a + b = 5
- a + b < 5
- a + b > 8
- If R1 = {(a, b) | b = 2a + 7, a ∈ R and – 5 ≤ a ≤ 5} is a relation. Find the domain and Range of R1.
- If R3 = {(a, a ) | a is a real number} is a relation. Find the domain and range of R3.
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