# NCERT Solutions for Class 12 Maths Maths Part-1 Chapter 1

## Relations and Functions Class 12

### Exercise 1.1 : Solutions of Questions on Page Number : 5

Q1 :

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

(i)Relation R in the set A = {1, 2, 3…13, 14} defined as

R = {(x, y): 3x - y = 0}

(ii) Relation R in the set N of natural numbers defined as

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

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

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

(iv) Relation R in the set Z of all integers defined as

R = {(x, y): x - y is as integer}

(v) 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}

Answer :

(i) A = {1, 2, 3 … 13, 14}

R = {(x, y): 3x - y = 0}

∴R = {(1, 3), (2, 6), (3, 9), (4, 12)}

R is not reflexive since (1, 1), (2, 2) … (14, 14) ∉ R.

Also, R is not symmetric as (1, 3) ∈R, but (3, 1) ∉ R. [3(3) - 1 ≠ 0]

Also, R is not transitive as (1, 3), (3, 9) ∈R, but (1, 9) ∉ R.

[3(1) - 9 ≠ 0]

Hence, R is neither reflexive, nor symmetric, nor transitive.

(ii) R = {(x, y): y = x + 5 and x < 4} = {(1, 6), (2, 7), (3, 8)}

It is seen that (1, 1) ∉ R.

∴R is not reflexive.

(1, 6) ∈R

But,

(6, 1) ∉ R.

∴R is not symmetric.

Now, since there is no pair in R such that (x, y) and (y, z) ∈R, then (x, z) cannot belong to R.

∴ R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.

(iii) A = {1, 2, 3, 4, 5, 6}

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

We know that any number (x) is divisible by itself. (x, x) ∈R

∴R is reflexive.

Now,

(2, 4) ∈R [as 4 is divisible by 2]

But,

(4, 2) ∉ R. [as 2 is not divisible by 4]

∴R is not symmetric.

Let (x, y), (y, z) ∈ R. Then, y is divisible by x and z is divisible by y.

z is divisible by x.

⇒ (x, z) ∈R

∴R is transitive.

Hence, R is reflexive and transitive but not symmetric.

(iv) R = {(x, y): x - y is an integer}

Now, for every xZ, (x, x) ∈R as x - x = 0 is an integer.

∴R is reflexive.

Now, for every x, yZ if (x, y) ∈ R, then x - y is an integer.

⇒ - (x - y) is also an integer.

⇒ (y - x) is an integer.

∴ (y, x) ∈ R

∴R is symmetric.

Now,

Let (x, y) and (y, z) ∈R, where x, y, zZ.

⇒ (x - y) and (y - z) are integers.

x - z = (x - y) + (y - z) is an integer.

∴ (x, z) ∈R

∴R is transitive.

Hence, R is reflexive, symmetric, and transitive.

(v) (a) R = {(x, y): x and y work at the same place} (x, x) ∈ R

∴ R is reflexive.

If (x, y) ∈ R, then x and y work at the same place.

y and x work at the same place.

⇒ (y, x) ∈ R.

∴R is symmetric.

Now, let (x, y), (y, z) ∈ R

x and y work at the same place and y and z work at the same place.

x and z work at the same place.

⇒ (x, z) ∈R

∴ R is transitive.

Hence, R is reflexive, symmetric, and transitive.

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

Clearly (x, x) ∈ R as x and x is the same human being.

∴ R is reflexive.

If (x, y) ∈R, then x and y live in the same locality.

y and x live in the same locality.

⇒ (y, x) ∈ R

∴R is symmetric.

Now, let (x, y) ∈ R and (y, z) ∈ R.

x and y live in the same locality and y and z live in the same locality.

x

Answer needs Correction? Click Here

Q2 :

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

R = {(a, b): ab2} is neither reflexive nor symmetric nor transitive.

Answer :

R = {(a, b): ab2}

It can be observed that ∴R is not reflexive.

Now, (1, 4) ∈ R as 1 < 42

But, 4 is not less than 12.

∴(4, 1) ∉ R

∴R is not symmetric.

Now,

(3, 2), (2, 1.5) ∈ R

(as 3 < 22 = 4 and 2 < (1.5)2 = 2.25)

But, 3 > (1.5)2 = 2.25

∴(3, 1.5) ∉ R

∴ R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.

Answer needs Correction? Click Here

Q3 :

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.

Answer :

Please Register/Login to get access to all solutions
Q4 :

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

Answer :

Please Register/Login to get access to all solutions
Q5 :

Check whether the relation R in R defined as R = {(a, b): ab3} is reflexive, symmetric or transitive.

Answer :

Please Register/Login to get access to all solutions
Q6 :

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.

Answer :

Please Register/Login to get access to all solutions
Q7 :

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.

Answer :

Please Register/Login to get access to all solutions
Q8 :

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by , 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}.

Answer :

Please Register/Login to get access to all solutions
Q9 :

Show that each of the relation R in the set , given by

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

Answer :

Please Register/Login to get access to all solutions
Q10 :

Given an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii) Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric.

(v) Symmetric and transitive but not reflexive.

Answer :

Please Register/Login to get access to all solutions
Q11 :

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.

Answer :

Please Register/Login to get access to all solutions
Q12 :

Show that the relation R defined in the set A of all triangles as R = {(T1, T2): 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 T1, T2 and T3 are related?

Answer :

Please Register/Login to get access to all solutions
Q13 :

Show that the relation R defined in the set A of all polygons as R = {(P1, P2): P1 and P2 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?

Answer :

Please Register/Login to get access to all solutions
Q14 :

Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): 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.

Answer :

Please Register/Login to get access to all solutions
Q15 :

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.

Answer :

Please Register/Login to get access to all solutions
Q16 :

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

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

Answer :

Please Register/Login to get access to all solutions

### Exercise 1.2 : Solutions of Questions on Page Number : 10

Q1 :

Show that the function f: R* Ã¢â€ ’ 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*?

Answer :

Please Register/Login to get access to all solutions
Q2 :

Check the injectivity and surjectivity of the following functions:

(i) f: NN given by f(x) = x2

(ii) f: ZZ given by f(x) = x2

(iii) f: RR given by f(x) = x2

(iv) f: NN given by f(x) = x3

(v) f: ZZ given by f(x) = x3

Answer :

Please Register/Login to get access to all solutions
Q3 :

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

Answer :

Please Register/Login to get access to all solutions
Q4 :

Show that the Modulus Function f: R Ã¢â€ ’ R given by , is neither one-one nor onto, where is x, if x is positive or 0 and is - x, if x is negative.

Answer :

Please Register/Login to get access to all solutions
Q5 :

Show that the Signum Function f: R Ã¢â€ ’ R, given by is neither one-one nor onto.

Answer :

Please Register/Login to get access to all solutions
Q6 :

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.

Answer :

Please Register/Login to get access to all solutions
Q7 :

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

(i) f: RR defined by f(x) = 3 - 4x

(ii) f: RR defined by f(x) = 1 + x2

Answer :

Please Register/Login to get access to all solutions
Q8 :

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

Answer :

Please Register/Login to get access to all solutions
Q9 :

Let f: N Ã¢â€ ’ N be defined by State whether the function f is bijective. Justify your answer.

Answer :

Please Register/Login to get access to all solutions
Q10 :

Let A = R - {3} and B = R - {1}. Consider the function f: A Ã¢â€ ’ B defined by . Is f one-one and onto? Justify your answer.

Answer :

Please Register/Login to get access to all solutions
Q11 :

Let f: RR 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

Answer :

Please Register/Login to get access to all solutions
Q12 :

Let f: RR 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

Answer :

Please Register/Login to get access to all solutions

### Exercise 1.3 : Solutions of Questions on Page Number : 18

Q1 :

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.

Answer :

Please Register/Login to get access to all solutions
Q2 :

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

Please Register/Login to get access to all solutions
Q3 :

Find gof and fog, if

(i) (ii) Answer :

Please Register/Login to get access to all solutions
Q4 :

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

Answer :

Please Register/Login to get access to all solutions
Q5 :

State with reason whether following functions have inverse

(i) f: {1, 2, 3, 4} → {10} with

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

(ii) g: {5, 6, 7, 8} → {1, 2, 3, 4} with

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

(iii) h: {2, 3, 4, 5} → {7, 9, 11, 13} with

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

Answer :

Please Register/Login to get access to all solutions
Q6 :

Show that f: [ - 1, 1] Ã¢â€ ’ R, given by is one-one. Find the inverse of the function f: [ - 1, 1] Ã¢â€ ’ Range f.

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

Answer :

Please Register/Login to get access to all solutions
Q7 :

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

Answer :

Please Register/Login to get access to all solutions
Q8 :

Consider f: R+ Ã¢â€ ’ [4, ) given by f(x) = x2+ 4. Show that fis invertible with the inverse f - 1of given f by , where R+is the set of all non-negative real numbers.

Answer :

Please Register/Login to get access to all solutions
Q9 :

Consider f: R+Ã¢â€ ’ [ - 5, ) given by f(x) = 9x2+ 6x - 5. Show that fis invertible with .

Answer :

Please Register/Login to get access to all solutions
Q10 :

Let f: XY be an invertible function. Show that f has unique inverse.

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

fog1(y) = IY(y) = fog2(y). Use one-one ness of f).

Answer :

Please Register/Login to get access to all solutions
Q11 :

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

Answer :

Please Register/Login to get access to all solutions
Q12 :

Let f: XY be an invertible function. Show that the inverse of f-1 is f, i.e.,

(f-1)-1 = f.

Answer :

Please Register/Login to get access to all solutions
Q13 :

If f: R Ã¢â€ ’ R be given by , then fof(x) is

(A) (B) x3 (C) x (D) (3 - x3)

Answer :

Please Register/Login to get access to all solutions
Q14 :

Let be a function defined as . The inverse of fis map g: Range (A) (B) (C) (D) Answer :

Please Register/Login to get access to all solutions

### Exercise 1.4 : Solutions of Questions on Page Number : 24

Q1 :

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.

(i) On Z+, define * by a * b = a - b

(ii) On Z+, define * by a * b = ab

(iii) On R, define * by a * b = ab2

(iv) On Z+, define * by a * b = |a - b|

(v) On Z+, define * by a * b = a

Answer :

Please Register/Login to get access to all solutions
Q2 :

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

(i) On Z, define a * b = a - b

(ii) On Q, define a * b = ab + 1

(iii) On Q, define a * b (iv) On Z+, define a * b = 2ab

(v) On Z+, define a * b = ab

(vi) On R - { - 1}, define Answer :

Please Register/Login to get access to all solutions
Q3 :

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

Answer :

Please Register/Login to get access to all solutions
Q4 :

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

Answer :

Please Register/Login to get access to all solutions
Q5 :

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

Answer :

Please Register/Login to get access to all solutions
Q6 :

Let * be the binary operation on N given by a * b = L.C.M. of a 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 N are invertible for the operation *?

Answer :

Please Register/Login to get access to all solutions
Q7 :

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.

Answer :

Please Register/Login to get access to all solutions
Q8 :

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?

Answer :

Please Register/Login to get access to all solutions
Q9 :

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

(i) a * b = a - b (ii) a * b = a2 + b2

(iii) a * b = a + ab (iv) a * b = (a - b)2

(v) (vi) a * b = ab2

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

Answer :

Please Register/Login to get access to all solutions
Q10 :

Find which of the operations given above has identity.

Answer :

Please Register/Login to get access to all solutions
Q11 :

Let A = N x 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.

Answer :

Please Register/Login to get access to all solutions
Q12 :

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

(i) For an arbitrary binary operation * on a set N, a * a = a a * N.

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

Answer :

Please Register/Login to get access to all solutions
Q13 :

Consider a binary operation * on N defined as a * b = 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?

Answer :

Please Register/Login to get access to all solutions

### Exercise Miscellaneous : Solutions of Questions on Page Number : 29

Q1 :

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

Answer :

Please Register/Login to get access to all solutions
Q2 :

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

Answer :

Please Register/Login to get access to all solutions
Q3 :

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.

Answer :

Please Register/Login to get access to all solutions
Q4 :

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.

Answer :

Please Register/Login to get access to all solutions
Q5 :

If f: RR is defined by f(x) = x2 - 3x + 2, find f(f(x)).

Answer :

Please Register/Login to get access to all solutions
Q6 :

Show that function f: R Ã¢â€ ’ {xR: - 1 < x < 1} defined by f(x) = , xR is one-one and onto function.

Answer :

Please Register/Login to get access to all solutions
Q7 :

Show that the function f: RR given by f(x) = x3 is injective.

Answer :

Please Register/Login to get access to all solutions
Q8 :

Give examples of two functions f: N Ã¢â€ ’ Z and g: Z Ã¢â€ ’ Z such that g o f is injective but g is not injective.

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

Answer :

Please Register/Login to get access to all solutions
Q9 :

Given examples of two functions f: N Ã¢â€ ’ N and g: N Ã¢â€ ’ N such that gof is onto but f is not onto.

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

Please Register/Login to get access to all solutions
Q10 :

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 AB. Is R an equivalence relation on P(X)? Justify you answer:

Answer :

Please Register/Login to get access to all solutions
Q11 :

Given a non-empty set X, consider the binary operation *: P(X) x P(X) → P(X) given by A * B = AB " A, B in 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*.

Answer :

Please Register/Login to get access to all solutions
Q12 :

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

Answer :

Please Register/Login to get access to all solutions
Q13 :

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.

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

Answer :

Please Register/Login to get access to all solutions
Q14 :

Consider the binary operations*: R ×R Ã¢â€ ’ and o: R × R Ã¢â€ ’ R defined as and a o b = a, "a, bR. Show that * is commutative but not associative, o is associative but not commutative. Further, show that "a, b, cR, 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.

Answer :

Please Register/Login to get access to all solutions
Q15 :

Given a non-empty set X, let *: P(X) 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).

Answer :

Please Register/Login to get access to all solutions
Q16 :

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.

Answer :

Please Register/Login to get access to all solutions
Q17 :

Let A = { - 1, 0, 1, 2}, B = { - 4, - 2, 0, 2} and f, g: A Ã¢â€ ’ B be functions defined by f(x) = x2 - x, x ∈ A and . Are f and g equal?

Justify your answer. (Hint: One may note that two function f: A Ã¢â€ ’ B and g: A Ã¢â€ ’ B such that f(a) = g(a) "aA, are called equal functions).

Answer :

Please Register/Login to get access to all solutions
Q18 :

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

Answer :

Please Register/Login to get access to all solutions
Q19 :

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

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

Answer :

Please Register/Login to get access to all solutions
Q20 :

Let f: R Ã¢â€ ’ R be the Signum Function defined as and g: R Ã¢â€ ’ 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]?

Answer :

Please Register/Login to get access to all solutions
Q21 :

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

(A) 10 (B) 16 (C) 20 (D) 8

Answer :

Please Register/Login to get access to all solutions

Maths Part-1 - Maths : CBSE NCERT Exercise Solutions for Class 12th for Relations and Functions ( Exercise 1.1, 1.2, 1.3, 1.4, miscellaneous ) will be available online in PDF book form soon. The solutions are absolutely Free. Soon you will be able to download the solutions.

Popular Articles
 Exercise 1.1 Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15 Question 16
 Exercise 1.2 Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12
 Exercise 1.3 Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14
 Exercise 1.4 Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13
 Exercise Miscellaneous Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15 Question 16 Question 17 Question 18 Question 19 Question 20 Question 21