Relations and Functions — Important Questions

The relation R on the set A = {1, 2, 3} given by R = {(1, 2), (2, 1)} is:

If f: R → R, f(x) = x² + 1, then f is:

Let f(x) = 2x + 3 and g(x) = x − 1. Then (fog)(2) equals:

The number of equivalence relations on the set {1, 2, 3} that contain (1, 2) is:

For a one-one function f: A → B, n(A) = 5 and n(B) = 7, the number of one-one functions from A to B is:

Define an equivalence relation and give one example.

Show that the function f: N → N defined by f(n) = 2n is one-one but not onto.

For f(x) = x + 1 and g(x) = x², find (fog)(x) and (gof)(x).

State the necessary and sufficient condition for a function f: A → B to be invertible.

On Z define R: aRb ⇔ a − b is divisible by 3. Show R is reflexive.

Show that the relation R on N × N defined by (a, b) R (c, d) ⇔ a + d = b + c is an equivalence relation.

Show that f: R − {−4/3} → R − {4/3} given by f(x) = (4x + 3) / (3x + 4) is a bijection.

Show that the relation R on Z given by aRb ⇔ a − b is divisible by 5 is an equivalence relation. Find the equivalence class of 0.

Let f: R → R, f(x) = x³ + 4. Show that f is bijective and find f⁻¹.

Let A = {1, 2, 3, ..., 9} and R be the relation in A × A defined by (a, b) R (c, d) iff a + d = b + c. Show that R is an equivalence relation. Also find equivalence class of (2, 5).

Differentiate between reflexive symmetric and transitive relations in tabular form with one example each.

1. Which of the following properties is satisfied by R?
   AReflexive

   BSymmetric

   CTransitive

   DAll of these
2. The equivalence class of 0, denoted [0], is best described as which of the listed sets?
   A{0, 5, 10}

   B{1, 6, 11}

   C{2, 7, 12}

   DAll sets above
3. Explain why R is an equivalence relation and describe the partition of Z it creates.

1. Since each student has a unique roll number and every roll number is used the function f is:
   AOne-one but not onto

   BOnto but not one-one

   CBoth one-one and onto

   DNeither
2. Because f is bijective:
   AInverse exists

   BInverse does not exist

   CInverse is many-valued

   DCannot decide
3. Explain in words why a bijection guarantees the existence of an inverse function.

1. The composite (g ∘ f)(x) is:
   A2x + 5

   B2x + 10

   C2(x + 5)

   Dx + 10
2. The composite (f ∘ g)(x) is:
   A2x + 5

   B2x + 10

   Cx + 5

   D4x + 5
3. Compute (g ∘ f)(3) and (f ∘ g)(3) and comment on whether composition is commutative.

Study the table of relations on set A = {1, 2, 3} and answer:

1. Which relation is an equivalence relation on A?
   AR1

   BR2

   CR3

   DR4
2. Which relation is symmetric and transitive but not reflexive?
   AR2

   BR3

   CR4

   D
3. Explain why R4 fails to be an equivalence relation despite being symmetric and transitive.

Study the function values and answer:

1. The composite (f ∘ g)(x) equals:
   A(x + 1)²

   Bx² + 1

   Cx²(x + 1)

   Dx + 1
2. Considering f: R → R defined by f(x) = x² the function is:
   AOne-one onto

   BOne-one not onto

   CMany-one onto

   DMany-one not onto
3. Explain whether f(x) = x², considered as a function from R to R, is one-one or onto.

For each relation R defined on A = {1, 2, 3} below, identify whether it is reflexive, symmetric or transitive.

Study the graphs of f(x) = x² and g(x) = x³ on R and answer:

1. As a function from R to R, f(x) = x² is:
   AOne-one and onto

   BOne-one but not onto

   CMany-one but onto

   DMany-one and not onto
2. As a function from R to R, g(x) = x³ is:
   AOne-one and onto

   BOne-one but not onto

   CMany-one but onto

   DMany-one and not onto
3. Use the horizontal-line test on each graph to justify your answers.