Functions and Equivalence Relations - One Line Questions
1.
What is the result of the composition f o g, where f(x) = x² and g(x) = x + 1? —
(x + 1)²
2.
What is the result of the composition g o f, where f(x) = x² and g(x) = x + 1? —
x² + 1
3.
If f(x) = 2x + 1, what is the inverse function f⁻¹(y)? —
(y - 1) / 2
4.
If R is an equivalence relation on A, and a R b, then the equivalence class of a, [a], and the equivalence class of b, [b], satisfy: —
[a] is equal to [b]
5.
For the relation R in the previous question, what are the equivalence classes? —
{[1, 2], [3, 4]}
6.
A function that is both injective and surjective is called: —
A bijective function
7.
If f: A -> B and g: B -> C are functions, what is the composition of f and g, denoted by g o f? —
A function from A to C defined by (g o f)(x) = g(f(x)).
8.
What is the identity function, id_A, on a set A? —
A function defined as id_A(x) = x for all x in A.
9.
Consider the relation 'has the same number of elements as' on the set of all finite sets. This is: —
An equivalence relation
10.
If a relation R on set A is reflexive and symmetric, it is called: —
A pre-order
11.
What is an equivalence relation? —
A relation that is reflexive, symmetric, and transitive.
12.
What is the fundamental definition of a function in mathematics? —
A relation where each input maps to exactly one output.
13.
What is the definition of a relation 'a is related to b' (a R b) on a set A? —
A subset of the Cartesian product A x A.
14.
What is the domain of the function f(x) = sqrt(x)? —
All non-negative real numbers.
15.
What is the range of the function f(x) = x²? —
All non-negative real numbers.
16.
A relation that is reflexive, antisymmetric, and transitive is called: —
A partial order relation
17.
Consider the relation '≤' on the set of real numbers. This is an example of: —
A partial order relation
18.
The relation R = {(x, y) | x and y are integers and x < y} on the set of integers is: —
Transitive but not reflexive or symmetric
19.
A function f: A -> B is called injective (or one-to-one) if: —
Different elements in A map to different elements in B.
20.
A function f: A -> B is called surjective (or onto) if: —
Every element in B is mapped to by at least one element in A.
21.
If f: A -> B and g: B -> A are inverse functions, then f(g(b)) = b for all b in B and g(f(a)) = a for all a in A. This means: —
f is bijective and g is bijective.
22.
If f: A -> B is a function, and g: B -> A is its inverse (g = f⁻¹), which property must hold? —
f o g = id_B and g o f = id_A
23.
A relation R on a set A is symmetric if: —
For all a, b in A, if (a, b) is in R, then (b, a) is in R.
24.
A relation R on a set A is transitive if: —
For all a, b, c in A, if (a, b) is in R and (b, c) is in R, then (a, c) is in R.
25.
A relation R on a set A is reflexive if: —
For all a in A, (a, a) is in R.
26.
A relation R on a set A is antisymmetric if: —
For all a, b in A, if (a, b) is in R and (b, a) is in R, then a = b.
27.
What does it mean for a function to be strictly increasing? —
If x₁ < x₂, then f(x₁) < f(x₂).
28.
If f is a bijective function from set A to set B, what can be said about the inverse function f⁻¹? —
It is a function from B to A.
29.
Consider the relation 'is parallel to' on the set of all lines in a plane. Is this relation an equivalence relation? —
Yes, it is reflexive, symmetric, and transitive.
30.
Let A = {1, 2, 3} and R = {(1,1), (2,2), (3,3), (1,2), (2,1)}. Is R an equivalence relation on A? —
No, it's not transitive.
31.
Consider the set of integers Z and the relation R defined as a R b if a - b is an even number. Is R an equivalence relation? —
Yes, it is an equivalence relation.
32.
Let A = {1, 2, 3, 4} and R = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (3,4), (4,3)}. Is R an equivalence relation? —
Yes, it is an equivalence relation.
33.
The relation R = {(x, y) | x and y are integers and x = y} on the set of integers is: —
An equivalence relation
34.
Which property is NOT required for a relation to be an equivalence relation? —
Antisymmetry
35.
Which of the following is a property of functions but not necessarily of relations? —
Uniqueness of output for each input
36.
Let A be a non-empty set. The relation R = {(a, a) | a ∈ A} on A is: —
An equivalence relation.
37.
The set of all equivalence classes of a set A under an equivalence relation R is called: —
The partition of A
38.
If R is an equivalence relation on a set A, the set of all elements related to an element 'a' in A is called: —
The equivalence class of 'a'
39.
What is the codomain of a function? —
The set of all values that the function's output can potentially take.
40.
In the context of the relation 'a R b if a - b is even' on integers, what is the equivalence class of the integer 3? —
The set of all odd integers.
41.
The range of a function is defined as: —
The set of all values in the codomain that are actually mapped to.
42.
In the context of functions, what does the domain refer to? —
The set of all possible input values.
43.
What is a key property of equivalence classes formed by an equivalence relation? —
They are always disjoint or identical.
44.
For a function f: A -> B, when does the inverse relation f⁻¹ exist as a function from B to A? —
When f is bijective.
45.
If f: A -> B is a function and g: B -> C is a function, and if f is not injective, can g o f be injective? —
No, never.
46.
If f: A -> B is a function and g: B -> C is a function, and if g is not surjective, can g o f be surjective? —
No, never.
47.
Consider the function f(x) = |x|. Is this function injective? —
No, because f(2) = 2 and f(-2) = 2.
48.
Consider the function f(x) = x³ - x. Is this function injective? —
No, for example, f(1) = 0 and f(-1) = 0.
49.
Consider the relation 'is congruent to' on the set of all triangles. Is this relation an equivalence relation? —
Yes, it is reflexive, symmetric, and transitive.