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NCERT solutions Mathematics Textbook for Class 12 Part 1 chapter 1 Relations and Functions

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Chapter 1 - Relations and Functions

Pages 5 - 7

Determine whether  the following relations are reflective, symmetric and transitive:

Relation R in the set  A  = {1, 2, 3...13, 14} defined as R = {(x,y):3x - y = 0}

Q 1.1 | Page 5 | view solution

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

Relation R in the set N of natural numbers defined as

R = {(xy): y = x + 5 and x < 4}

Q 1.2 | Page 5 | view solution

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

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

R = {(xy): y is divisible by x}

Q 1.3 | Page 5 | view solution

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

Relation R in the set Z of all integers defined as

R = {(xy): x − y is as integer}

Q 1.4 | Page 5 | view solution

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

(a) R = {(xy): and y work at the same place}

(b) R = {(xy): x and y live in the same locality}

(c) R = {(xy): is exactly 7 cm taller than y}

(d) R = {(xy): x is wife of y}

(e) R = {(xy): x is father of y}

Q 1.5 | Page 5 | view solution

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

R = {(ab): a ≤ b2} is neither reflexive nor symmetric nor transitive.

Q 2 | Page 5 | view solution

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(ab): b = a + 1} is reflexive, symmetric or transitive.

Q 3 | Page 5 | view solution

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

Q 4 | Page 5 | view solution

Check whether the relation R in R defined as R = {(ab): a ≤ b3} is reflexive, symmetric or transitive.

Q 5 | Page 5 | view solution

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.

Q 6 | Page 6 | view solution

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

Q 7 | Page 6 | view solution

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}.

Q 8 | Page 6 | view solution

Show that each of the relation R in the set A= {x in 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.

Q 9.1 | Page 6 | view solution

Show that each of the relation R in the set `A = {x =Z: 0 <= <= 12 }`  given by `R = {(a,b) : a= b}

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

Q 9.2 | Page 6 | view solution

Given an example of a relation. Which is Symmetric but neither reflexive nor transitive.

Q 10.1 | Page 6 | view solution

Given an example of a relation. Which is Transitive but neither reflexive nor symmetric.

Q 10.2 | Page 6 | view solution

Given an example of a relation. Which is  Reflexive and symmetric but not transitive.

Q 10.3 | Page 6 | view solution

Given an example of a relation. Which is Reflexive and transitive but not symmetric.

Q 10.4 | Page 6 | view solution

Given an example of a relation. Which is Symmetric and transitive but not reflexive.

Q 10.5 | Page 6 | view solution

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 point related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

Q 11 | Page 6 | view solution

Show that the relation R defined in the set A of all triangles as R = {(T1T2): T1 is similar to T2}, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1T2and T3 are related?

Q 12 | Page 6 | view solution

Show that the relation R defined in the set A of all polygons as R = {(P1P2): P1 and P2have 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?

Q 13 | Page 6 | view solution

Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

Q 14 | Page 6 | view solution

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.

(A) R is reflexive and symmetric but not transitive.

(B) R is reflexive and transitive but not symmetric.

(C) R is symmetric and transitive but not reflexive.

(D) R is an equivalence relation.

Q 15 | Page 7 | view solution

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

(A) (2, 4) ∈ R (B) (3, 8) ∈R (C) (6, 8) ∈R (D) (8, 7) ∈ R

Q 16 | Page 7 | view solution

Pages 10 - 11

Show that the function fR* → R* defined by `f(x) = 1/x` 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?

Q 1 | Page 10 | view solution

Check the injectivity and surjectivity of the following functions:

fN → N given by f(x) = x2

Q 2.1 | Page 10 | view solution

Check the injectivity and surjectivity of the given functions: fZ → Z given by f(x) = x2

Q 2.2 | Page 10 | view solution

Check the injectivity and surjectivity of the given functions:  fR → R given by f(x) = x2

Q 2.3 | Page 10 | view solution

Check the injectivity and surjectivity of the following functions: f→ N given by f(x) = x3

Q 2.4 | Page 10 | view solution

Check the injectivity and surjectivity of the following functions: fZ → Z given by f(x) = x3

Q 2.5 | Page 10 | view solution

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

Q 3 | Page 10 | view solution

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

Q 4 | Page 11 | view solution

Show that the Signum Function fR → R, given by

 

 

Q 5 | Page 11 | view solution

Let A = {1, 2, 3}, = {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.

Q 6 | Page 11 | view solution

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

f→ R defined by f(x) = 3 − 4x

Q 7.1 | Page 11 | view solution

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

f→ R defined by f(x) = 1 + x2

Q 7.2 | Page 11 | view solution

Let A and B be sets. Show that fA × B → × A such that (ab) = (ba) is bijective function.

Q 8 | Page 11 | view solution

Let fN → N be defined by f(n) = `{((n+1)/2, "if n is odd"),(,"   for all n ∈ N"), (n/2, if "n is even"):}`

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

Q 9 | Page 11 | view solution

Let A = R − {3} and B = R − {1}. Consider the function f: A → B defined by `f(x) = ((x- 2)/(x -3))`. Is f one-one and onto? Justify your answer.

Q 10 | Page 11 | view solution

Let fR → R be defined as f(x) = x4. Choose the correct answer.

(A) f is one-one onto (B) f is many-one onto

(C) f is one-one but not onto (D) f is neither one-one nor onto

Q 11 | Page 11 | view solution

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

(A) f is one-one onto (B) f is many-one onto

(C) f is one-one but not onto (D) f is neither one-one nor onto

Q 12 | Page 11 | view solution

Pages 18 - 19

Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by = {(1, 2), (3, 5), (4, 1)} and = {(1, 3), (2, 3), (5, 1)}. Write down gof.

Q 1 | Page 18 | view solution

Let fg and h be functions from to R. Show that

`(f + g)oh = foh + goh`

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

Q 2 | Page 18 | view solution

Find gof and fog, if  f(x) = |x| and g(x) = |5x - 2|

Q 3.1 | Page 18 | view solution

Find goand fog, if `f(x) = 8x^3` and `g(x) = x^(1/3)`

 

Q 3.2 | Page 18 | view solution

if f(x) = `(4x + 3)/(6x - 4), x != 2/3` show that fof(x) = x, for all x != 2/3 . What is the inverse of f?

Q 4 | Page 18 | view solution

State with reason whether following functions have inverse

f: {1, 2, 3, 4} → {10} with

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

Q 5.1 | Page 18 | view solution

State with reason whether following functions have inverse g: {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}

Q 5.3 | Page 18 | view solution

State with reason whether following functions have inverse h: {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}

Q 5.3 | Page 18 | view solution

Show that f: [−1, 1] → R, given by f(x) = `x/(x + 2)`  is one-one. Find the inverse of the function f: [−1, 1] → Range f.

(Hint: For y in Range f, y = `f(x) = x/(x +2)` for some x in [-1, 1] ie x = `2y/(1-y)`

Q 6 | Page 18 | view solution

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

Q 7 | Page 18 | view solution

Consider fR→ [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with the inverse f−1 of given by `f^(-1) (y) = sqrt(y - 4)` where R+ is the set of all non-negative real numbers.

Q 8 | Page 18 | view solution

Consider fR+ → [−5, ∞) given by f(x) = 9x2 + 6x − 5. Show that f is invertible with `f^(-1)(y) = ((sqrt(y +6) - 1)/3)`

Q 9 | Page 19 | view solution

Let fX → 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 ∈ Yfog1(y) = IY(y) = fog2(y). Use one-one ness of f).

Q 10 | Page 19 | view solution

Consider f: {1, 2, 3} → {abc} given by f(1) = af(2) = b and f(3) = c. Find f−1 and show that (f−1)−1 = f.

Q 11 | Page 19 | view solution

Let fX → Y be an invertible function. Show that the inverse of f−1 is f, i.e., (f−1)−1 = f.

Q 12 | Page 19 | view solution

If f→ be given by `f(x) = (3 - x^3)^(1/3)` , then fof(x) is 

(A) `1/(x^3)`

(B) x3

(C) x

(D) (3 − x3)

Q 13 | Page 19 | view solution

Let `f:R - {-4/3} -> R` be a function defined as `f(x) = (4x)/(3x + 4)`. The inverse of f is map g Range `f -> R -{- 4/3}`

(A) `g(y) = (3y)/(3-4y)`

(B) `g(y) = (4y)/(4 - 3y)`

(C) `g(y) = (4y)/(3 - 4y)` 

(D) `g(y) = (3y)/(4 - 3y)`

Q 14 | Page 19 | view solution

Pages 24 - 26

Determine whether or not 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

Q 1.1 | Page 24 | view solution

Determine whether or not 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 ab

Q 1.2 | Page 24 | view solution

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 R, define * by ab2

Q 1.3 | Page 24 | view solution

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 = |− b|

Q 1.4 | Page 24 | view solution

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

Q 1.5 | Page 24 | view solution

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

On Z, define − b

Q 2.1 | Page 24 | view solution

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

On Q, define ab + 1

Q 2.2 | Page 24 | view solution

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

On Q, define a * b  = `(ab)/2`

Q 2.3 | Page 24 | view solution

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

On Z+, define = 2ab

Q 2.4 | Page 24 | view solution

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

On Z+, define ab

Q 2.5 | Page 24 | view solution

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

On − {−1}, define `a*b = a/(b+1)`

Q 2.6 | Page 24 | view solution

Consider the binary operation ∨ on the set {1, 2, 3, 4, 5} defined by = min {ab}. Write the operation table of the operation∨.

Q 3 | Page 24 | view solution

Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table.

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

(ii) Is * commutative?

(iii) Compute (2 * 3) * (4 * 5).

(Hint: use the following table)

* 1 2 3 4 5
1 1 1 1 1 1
2 1 2 1 2 1
3 1 1 3 1 1
4 1 2 1 4 1
5 1 1 1 1 5
Q 4 | Page 25 | view solution

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

Q 5 | Page 25 | view solution

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

(i) 5 * 7, 20 * 16

(ii) Is * commutative?

(iii) Is * associative?

(iv) Find the identity of * in N

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

Q 6 | Page 25 | view solution

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

Q 8 | Page 25 | view solution

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

(i) − 

(ii) a2 + b2

(iii) ab 

(iv) = (− b)2

(v) a * b = ab/4

(vi) ab2

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

Q 9 | Page 25 | view solution

Find which of the operations given above has identity.

Q 10 | Page 25 | view solution

Let A = × and * be the binary operation on A defined by  (ab) * (cd) = (cd)

Show that * is commutative and associative. Find the identity element for * on A, if any.

Q 11 | Page 25 | view solution

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

For an arbitrary binary operation * on a set N= ∀  N.

Q 12.1 | Page 26 | view solution

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

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

Q 12.2 | Page 26 | view solution

Consider a binary operation * on defined as a3 + b3. Choose the correct answer.

(A) Is * both associative and commutative?

(B) Is * commutative but not associative?

(C) Is * associative but not commutative?

(D) Is * neither commutative nor associative?

Q 13 | Page 26 | view solution

Pages 29 - 31

Let fR → be defined as f(x) = 10x + 7. Find the function gR → R such that g o f = f o = 1R.

Q 1 | Page 29 | view solution

Let f: W → W be defined as f(n) = n − 1, if 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.

Q 2 | Page 29 | view solution

If f→ R is defined by f(x) = x2 − 3+ 2, find f(f(x)).

Q 3 | Page 29 | view solution

Show that function fR → {x ∈ R: −1 < x < 1} defined by f(x) =`x/(1 + |x|)` , R is one-one and onto function.

Q 4 | Page 29 | view solution

Show that the function fR → R given by f(x) = x3 is injective.

Q 5 | Page 29 | view solution

Give examples of two functions fN → Z and gZ → Z such that g o f is injective but gis not injective.

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

Q 6 | Page 29 | view solution

Given examples of two functions fN → N and gN → N such that gof is onto but is not onto.

(Hint: Consider f(x) = x + 1 and `g(x) = {(x-1, ifx >1),(1, if x = 1):}`

Q 7 | Page 29 | view solution

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 AB in P(X), ARB if and only if A ⊂ B. Is R an equivalence relation on P(X)? Justify you answer:

Q 8 | Page 29 | view solution

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

Q 9 | Page 30 | view solution

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

Q 10 | Page 30 | view solution

Let S = {abc} 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)} 

Q 11.1 | Page 30 | view solution

Let S = {abc} and T = {1, 2, 3}. Find F−1 of the following functions F from S to T, if it exists.

F = {(a, 2), (b, 1), (c, 1)}

Q 11.2 | Page 30 | view solution

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

Q 12 | Page 30 | view solution

Given a non-empty set X, let *: P(X) × P(X) → P(X) be defined as A * B = (A − B) ∪ (B −A), &mnForE; AB ∈ 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) = Aand (A − A) ∪ (A − A) = A * A = Φ).

Q 13 | Page 30 | view solution

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

a * b = `{(a+b, "if a+b < 6"), (a + b - 6, if a +b >= 6):}`

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.

Q 14 | Page 30 | view solution

Let A = {−1, 0, 1, 2}, B = {−4, −2, 0, 2} and fgA → B be functions defined by f(x) = x2 − xx ∈ A and g(x) = `2|x - 1/2|- 1, x in A`. Are f and g equal?

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

Q 15 | Page 30 | view solution

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

(A) 1 (B) 2 (C) 3 (D) 4

Q 16 | Page 30 | view solution

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

(A) 1

(B) 2

(C) 3

(D) 4

Q 16 | Page 30 | view solution

Let fR → R be the Signum Function defined as

f(x) = `{(1,x>0), (0, x =0),(-1, x< 0):}`

and gR → 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]?

Q 18 | Page 31 | view solution

Number of binary operations on the set {ab} are

(A) 10

(B) 16

(C) 20

(D) 8

Q 19 | Page 31 | view solution

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