Page 55 - Algebra

f : X → Y is denoted by y = f(x)
For example, x > 2, f(x) = 2x + 1 and for -1 ≤ x ≤ 1, f(x) = x – 2 The value of f(3) + f(1) = (2(3) + 1) + (1 – 2) = 7 – 1 = 6
3.2. Relation, Domain and Range
Relation
P and Q are two non-empty sets such that P{𝑎, 𝑏}, Q {1, 4, 5}, then P × Q = {(𝑎, 1), (𝑎, 4), (𝑎, 5), (b, 1), (b, 4), (b, 5)},
Q × P = {(1, 𝑎), (4, 𝑎), (5, 𝑎), (1, b), (4, b), (5, b)}
This is the Cartesian product of two non-empty sets.
Therefore, if P and Q are two non-empty sets, then the subset of P × Q is also the relation from P to Q.
For example,
P = {𝑎, 𝑏}, Q = {1, 4, 5}
Then {(𝑎, 1), (𝑎, 4), (𝑏, 5)} is a relation in P × Q. There will be many more relations in P × Q.
Domain and Range of Relations
‘R’ is the relation from set P to set Q.
REMEMBER:
• The domain is the set of all the first components of the ordered pairs that belongs to R. Domain (R) = {𝑝: (𝑝, 𝑞) 𝜖 𝑅}
• The range is the set of all the second components of the ordered pairs that belongs to R. Range (R) = {𝑞: (𝑝, 𝑞) 𝜖 𝑅}
Worked Example
P × Q P = {1, 2, 3}, Q = {3, 2, 6}
R = {(1, 3), (1, 2), (1, 6), (2, 3), (2, 2), (2, 6), (3,3), (3, 2), (3, 6)} Domain (R) = {1, 2, 3}
Range (R) = {2, 3, 6}
Find the relation
, range and domain when
Solution:
Inverse of a relation
For any relation R, the inverse of a relation is the interchanging of the first and second components of
-1 the given ordered pair. It’s denoted by R .
R = {(1, 3), (1, 2), (1, 6)} R-1 = {(3, 1), (2, 1), (6, 1)}
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Algebra I & II