Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation R in the set A = {1, 2, 3, ..., 13, 14} defined as

R = {(x, y) : 3x – y = 0}

Relation R in the set N of natural numbers defined as

R = {(x, y) : y = x + 5 and x < 4}

Relation R in the set A = {1, 2, 3, 4, 5, 6} as

R = {(x, y) : y is divisible by x}

Relation R in the set Z of all integers defined as

R = {(x, y) : x – y is an integer}

Relation R in the set A of human beings in a town at a particular time given by

A. R = {(x, y) : x and y work at the same place}

B. R = {(x, y) : x and y live in the same locality}

C. R = {(x, y) : x is exactly 7 cm taller than y}

D. R = {(x, y) : x is wife of y}

E. R = {(x, y) : x is father of y}

Show that the relation R in the set R of real numbers, defined as

R = {(a, b) : a ≤ b^{2}} is neither reflexive nor symmetric nor transitive.

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as

R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.

Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.

Check whether the relation R in R defined by R = {(a, b) : a ≤ b^{3}} is reflexive, symmetric or transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y): x and y have same number of pages} is an equivalence relation.

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by

R = {(a, b) : |a – b| is a multiple of 4}

is an equivalence relation. Find the set of all elements related to 1 in each case.

R = {(a, b) : a = b}

Give an example of a relation. Which is

Symmetric but neither reflexive nor transitive.

Transitive but neither reflexive nor symmetric.

Reflexive and symmetric but not transitive.

Reflexive and transitive but not symmetric.

Symmetric and transitive but not reflexive.

Show that the relation R in the set A of points in a plane given by

R = {(P, Q) : distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

Show that the relation R defined in the set A of all triangles as R = {(T_{1}, T_{2}): T_{1} is similar to T_{2}}, is equivalence relation. Consider three right angle triangles T_{1} with sides 3, 4, 5, T_{2} with sides 5, T_{2}, 13 and T_{3} with sides 6, 8, 10. Which triangles among T_{1}, T_{2} and T_{3} are related?

Show that the relation R defined in the set A of all polygons as R = {(P_{1}, P_{2}): P_{1} and P_{2} have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Let L be the set of all lines in XY plane and R be the relation in L defined as

R = {(L_{1}, L_{2}): L_{1} is parallel to L_{2}}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.

Let R be the relation in the set N given by R = {(a, b): a = b – 2, b > 6}. Choose the correct answer.

Show that the function f : R* → R* defined by is one-one and onto, where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R*?

Check the injectivity and surjectivity of the following functions:

f : N → N given by f (x) = x2

f : Z → Z given by f (x) = x^{2}

f : R → R given by f (x) = x^{2}

f : N → N given by f (x) = x^{3}

f : Z → Z given by f (x) = x^{3}

Prove that the Greatest Integer Function f : R → R, given by f (x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Show that the Modulus Function f : R → R, given by f (x) = |x|, is neither one-one nor onto, where |x| is x, if x is positive or 0 and |x| is – x, if x is negative.

Show that the Signum Function f : R → R, given by

is neither one-one nor onto.

Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

f : R → R defined by f (x) = 3 – 4x

f : R → R defined by f (x) = 1 + x^{2}

Let A and B be sets. Show that f : A × B → B × A such that f (a, b) = (b, a) is bijective function.

Let f : N → N be defined by

State whether the function f is bijective. Justify your answer.

Let A = R – {3} and B = R – {1}. Consider the function f: A → B defined by . Is f one-one and onto? Justify your answer.

Let f: R → R be defined as f(x) = x^{4}. Choose the correct answer.

Let f: R → R be defined as f (x) = 3x. Choose the correct answer.

Let f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3} be given by

f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.

Let f, g and h be functions from R to R. Show that

(f + g)oh = foh + goh

(f . g)oh = (foh) . (goh)

Find gof and fog, if

(i) f (x) = | x | and g(x) = | 5x – 2 |

If , show that fof(x) = x, for all . What is the inverse of f?

State with reason whether following functions have inverse

f : {1, 2, 3, 4} → {10} with

f = {(1, 10), (2, 10), (3, 10), (4, 10)}

g : {5, 6, 7, 8} → {1, 2, 3, 4} with

g = {(5, 4), (6, 3), (7, 4), (8, 2)}

h: {2, 3, 4, 5} → {7, 9, 11, 13} with

h = {(2, 7), (3, 9), (4, 11), (5, 13)}

Show that f : [–1, 1] → R, given by is one-one. Find the inverse of the function f : [–1, 1] → Range f.

(Hint: For y ∈ Range f, y = , for some x in [–1, 1], i.e., )

Consider f : R → R given by f (x) = 4x + 3. Show that f is invertible. Find the inverse of f.

Consider f : R_{+}→ [4, ∞) given by f (x) = x^{2} + 4. Show that f is invertible with the inverse f –1 of f given by , where R+ is the set of all non-negative real numbers.

Consider f: R+ → [–5, ∞) given by f(x) = 9x^{2} + 6x – 5. Show that f is invertible with

Let f: X → Y be an invertible function. Show that f has unique inverse.

(Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y,

fog_{1}(y) = 1_{Y}(y) = fog_{2}(y). Use one-one ness of f).

Consider f: {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f^{–}^{1} and show that (f^{–1})^{–1} = f.

Let f: X → Y be an invertible function. Show that the inverse of f^{–1} is f, i.e., (f^{–1})^{–1} = f.

If f : R → R be given by , then fof (x) is

Let R be a function defined as . The inverse of f is the map g : Range given by

Determine whether or not each of the definition of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this.

On Z^{+}, define ∗ by a ∗ b = a – b

On Z^{+}, define ∗ by a ∗ b = ab

On R, define ∗ by a ∗ b = ab^{2}

On Z^{+}, define ∗ by a ∗ b = |a – b|

On Z^{+} define ∗ by a ∗ b = a

For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

On Z, define a ∗ b = a – b

On Q, define a ∗ b = ab + 1

On Q, define a ∗ b =

On Z^{+}, define a ∗ b = 2^{ab}

On Z^{+}, define a ∗ b = a^{b}

On R – {–1}, define a ∗ b =

consider the binary operation ∧ on the set {1, 2, 3, 4, 5} defined by a ∧ b = min {a, b}. Write the operation table of the operation ∧.

Consider a binary operation ∗ on the set {1, 2, 3, 4, 5} given by the following multiplication table (Table 1.2).

(i) Compute (2 ∗ 3) ∗ 4 and 2 ∗ (3 ∗ 4)

(ii) Is ∗ commutative?

(iii) Compute (2 ∗ 3) ∗ (4 ∗ 5).

(Hint: use the following table)

Table 1.2

Let ∗′ be the binary operation on the set {1, 2, 3, 4, 5} defined by a ∗′ b = H.C.F. of a and b. Is the operation ∗′ same as the operation ∗ defined in Exercise 4 above? Justify your answer.

Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find

(i) 5 ∗ 7, 20 ∗ 16

(iii) Is ∗ associative?

(iv) Find the identity of ∗ in N

(v) Which elements of N are invertible for the operation ∗?

Is ∗ defined on the set {1, 2, 3, 4, 5} by a ∗ b = L.C.M. of a and b a binary operation? Justify your answer.

Let ∗ be the binary operation on N defined by a ∗ b = H.C.F. of a and b. Is ∗ commutative? Is ∗ associative? Does there exist identity for this binary operation on N?

Let ∗ be a binary operation on the set Q of rational numbers as follows:

a ∗ b = a – b

Find which of the binary operations are commutative and which are associative.

a ∗ b = a^{2} + b^{2}

a ∗ b = a + ab

a ∗ b = (a – b)^{2}

a ∗ b =

a ∗ b = ab^{2}

Find which of the operations given above has identity.

Let A = N × N and ∗ be the binary operation on A defined by

(a, b) ∗ (c, d) = (a + c, b + d)

Show that ∗ is commutative and associative. Find the identity element for ∗ on

A, if any.

State whether the following statements are true or false. Justify.

For an arbitrary binary operation ∗ on a set N, a ∗ a = a ∀a ∈ N.

If ∗ is a commutative binary operation on N, then a ∗ (b ∗ c) = (c ∗ b) ∗ a

Consider a binary operation ∗ on N defined as a ∗ b = a^{3} + b^{3}. Choose the correct answer.

Let f: R → R be defined as f (x) = 10x + 7. Find the function g : R → R such that g o f = f o g = 1_{R}.

Let f : W → W be defined as f (n) = n – 1, if n is odd and f (n) = n + 1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers.

If f : R → R is defined by f(x) = x^{2} – 3x + 2, find f (f (x)).

Show that the function f: R → {x ∈ R : – 1 < x < 1} defined by is one-one and onto function.

Show that the function f : R → R given by f (x) = x^{3} is injective.

Give examples of two functions f: N → Z and g: Z → Z such that g o f is injective but g is not injective.

(Hint: Consider f (x) = x and g(x) = |x|).

Give examples of two functions f: N → N and g : N → N such that g o f is onto but f is not onto.

(Hint: Consider f (x) = x + 1 and

Given a non-empty set X, consider P(X) which is the set of all subsets of X.

Define the relation R in P(X) as follows:

For subsets A, B in P(X), ARB if and only if A ⊂ B. Is R an equivalence relation on P(X)? Justify your answer.

Given a non-empty set X, consider the binary operation ∗: P(X) × P(X) → P(X) given by A ∗ B = A ∩ B ∀ A, B in P(X), where P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation ∗.

Find the number of all onto functions from the set {1, 2, 3, ..., n} to itself.

Let S = {a, b, c} and T = {1, 2, 3}. Find F^{–1} of the following functions F from S to T, if it exists.

F = {(a, 3), (b, 2), (c, 1)}

F = {(a, 2), (b, 1), (c, 1)}

Consider the binary operations ∗: R × R → R and o: R × R → R defined as a ∗b = |a – b| and a o b = a, ∀ a, b ∈ R. Show that ∗ is commutative but not associative, o is associative but not commutative. Further, show that ∀a, b, c ∈ R, a ∗ (b o c) = (a ∗ b) o (a ∗ c). [If it is so, we say that the operation ∗ distributes over the operation o]. Does o distribute over ∗? Justify your answer.

Given a non-empty set X, let ∗: P(X) × P(X) → P(X) be defined as A * B = (A – B) ∪ (B – A), ∀A, B ∈ P(X). Show that the empty set φ is the identity for the operation ∗ and all the elements A of P(X) are invertible with

A^{–1} = A.

(Hint: (A – φ) ∪ (φ – A) = A and (A – A) ∪ (A – A) = A ∗ A = φ).

Define a binary operation ∗ on the set {0, 1, 2, 3, 4, 5} as

Show that zero is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 – a being the inverse of a.

Let A = {–1, 0, 1, 2}, B = {–4, –2, 0, 2} and f, g: A → B be functions defined by f (x) = x^{2} – x, x ∈ A and , x ∈ A. Are f and g equal?

Justify your answer. (Hint: One may note that two functions f: A → B and g: A → B such that f (a) = g(a) ∀ a ∈ A, are called equal functions).

Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is

Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is

Let f: R → R be the Signum Function defined as

and g: R → R be the Greatest Integer Function given by g(x) = [x], where [x] is greatest integer less than or equal to x. Then, does fog and gof coincide in (0, 1]?

Number of binary operations on the set {a, b} are