Formalizing relations and functions practice
Here are some practice problems on formalizing relations and functions:
1. Given the relation R = {(1, 2), (3, 4), (5, 6)}, determine if it is reflexive, symmetric, and transitive.
2. Consider the function f: A → B, where A = {1, 2, 3, 4} and B = {a, b, c, d}, defined by the following mapping: f(1) = b, f(2) = c, f(3) = a, f(4) = c. Determine the range of the function.
3. Given the relations R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)} and S = {(2, 4), (4, 6)}, find the composition R ∘ S.
4. Determine if the relation R = {(x, y) ∈ ℝ × ℝ | x + y = 4} is a function.
5. Given the function f: A → B, where A = {1, 2, 3, 4} and B = {a, b, c, d}, defined by the following mapping: f(1) = b, f(2) = c, f(3) = a, f(4) = c. Determine if the function is injective (one-to-one) and surjective (onto).
6. Consider the function f: ℝ → ℝ, defined by f(x) = x^2. Determine if the function is injective (one-to-one) and surjective (onto).
7. Given the relation R = {(x, y) ∈ ℝ × ℝ | x^2 + y^2 = 1}, determine if it is reflexive, symmetric, and transitive.
8. Given the relations R = {(1, 2), (2, 3), (3, 4), (4, 1)} and S = {(2, 3), (3, 4), (4, 1)}, find the composition R ∘ S.
9. Determine if the relation R = {(x, y) ∈ ℝ × ℝ | x^2 = y} is a function.
10. Given the function f: A → B, where A = {1, 2, 3} and B = {a, b, c}, defined by the following mapping: f(1) = b, f(2) = c, f(3) = a. Determine if the function is injective (one-to-one) and surjective (onto).
I hope these practice problems help you reinforce your understanding of formalizing relations and functions!