A relation in mathematics is a rule that connects elements of one set to elements of another set (or the same set). It is represented as a set of ordered pairs, where the first element belongs to the domain and the second element belongs to the codomain (or range of possible values).
Question 1: Find the inverse relation of R = {(1, 3), (2, 4), (3, 5)}
Solution:
Inverse relation is defined as R-1 = {(y, x): (x, y) ∈ R}
Given,
R = {(1, 3), (2, 4), (3, 5)}Inverse relation of R is R-1 then,
R-1 = {(3, 1), (4, 2), (5, 3)}
Question 2: Is the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)} on set A = {1, 2, 3} an equivalence relation?
Solution:
No, it's not an equivalence relation. It's reflexive and symmetric, but not transitive.
Question 3: Find the domain and range of the relation R = {(1, 2), (2, 3), (3, 4), (4, 5)}.
Solution:
Domain = {1, 2, 3, 4}, Range = {2, 3, 4, 5}
Question 4: Is the relation R = {(x, y) | x² + y² = 25} on the set of real numbers a function?
Solution:
No, it's not a function. For example, (0, 5) and (0, -5) are both in the relation.
Question 5: Find the composition R ∘ S for R = {(1, 2), (2, 3), (3, 4)} and S = {(1, 1), (2, 2), (3, 3), (4, 4)}.
Solution:
R ∘ S = {(1, 2), (2, 3), (3, 4)}
Question 6: Find the reflexive closure of R = {(1, 2), (2, 3), (3, 1)} on set A = {1, 2, 3}.
Solution:
Reflexive closure = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (3, 1)}
Practice Questions
Question 1: Find the inverse relation of R = {(2, 5), (3, 6), (4, 7)}.
Question 2: Determine whether the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} on the set A = {1, 2, 3} is reflexive, symmetric, and transitive.
Question 3: Find the domain and range of R = {(2, 4), (3, 9), (4, 16), (5, 25)}.
Question 4: Find the composition of relations R∘S, where R = {(1, 2), (2, 3), (3, 1)} and S = {(1, 3), (2, 1), (3, 2)}.
Question 5: Find the reflexive closure of R = {(a, b), (b, c)} on the set A = {a, b, c}.
