EXERCISE - A
2) Which of the following are relation from B to A, where A={a,b,c,d} and B={x, y,z} ?
i) {(z,x),(z,y),(x,a)}
ii) {(z,a),(z,b),(z,c),(z,d)}
iii) {(x,a),(y,b)}
iv) {(b,y),(c,z),(a,x)}
v) {x,d),(y,c),(z,b)}
3) In each of the following state which of the ordered pairs belong to the given relation ?
i) {(x,y): x > y+ 5} ; (1,0),(8,2),(0,1),(2,8),(9,3), (10,7), (123,4)
ii) {(x,y): xy= 12} ; (3,4),(4,3),(12,0),(0,12),(12,1),(6, 2),(7,5).
iii) {(x,y: y= (x +3)/(x -1), x ∉ 1}; (0,1),(2,5),(5,2),(3,3),(7,5),(7,5/3).
4) Let N be the set of natural numbers. Describe the following relations in words, giving their domain and the range.
a) {(2,1),(4,2),(10,5),(18,9),(20,10)}
b) {(3,1),(6,2),(15,5)}
c) {(1,4),(5,16),(7,22),(12,37)}.
5) Z is set of integers. Describe the following relation in set builder form, given its domain and range.
{(0,-7),(2,-5),(4,-3),(-13,-20),....}
6) Write down the domain and range of the relation (x,y): x =3y and x and y are natural numbers less than 10.
7) Determine the domain and range of the relation R.
a) R={(x +1),(x +5): x ∈ {0,1,2,3,4,5}
b) R = {(x, x³) : x is a prime number less 10}
8) Given A={-2,-1,0,1,2}, list the ordered pairs determined by each of the following relations applied on A:
i) R₁= 'is less than '
ii) R₂ = " is the square of".
iii) R₃ = "is the additive inverse of".
iv) R₄ = 'is equal to '.
9) Given A= {2,3,4,5,6}. List the element of each of the following relations:
i) {(x,y) ∈ A x A : x = y}.
ii) {(x,y) ∈ A x A : x > y, x/y ∉ W}.
iii) {(x,y) ∈ A x A : x is a divisor of y and x ≠ y}
EXERCISE - B
1) Consider the following properties of relations : Symmetric(S), Transitive(T), Reflexive (R), Equivalence (E), None of these (N).
State which property/properties are satisfied by each of the following relations :
(Give the answers in terms of S, T, R, E and N).
a) "is greater than" for the set of real numbers.
b) "is the cube of" for the set of all real numbers.
c) " is the sister of" for a set of children.
d) " is similar to" for the set of triangles.
e) " is parapendicular to" for a set of co-planar lines.
2) Write down a relation which is
a) Only transitive
b) Only symmetric .
c) Only reflexive and transitive.
d) Only symmetric and reflexive.
3) Prove that if A is the set of the members of a family and R means " is brother of" then it is a transitive relation.
4) 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.
5) Let R be the relation in the set =1,2,3,4} given by {(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 simmetric.
c) R is symmetric and transitive but not reflective.
d) R is an equivalence relation.
6) Show that the relation R in the set A of real numbers defined as R= {(a,b): a ≤ b} is Reflexive and transitive but not symmetric.
7) Show that the relation R in the set A of real numberr defined as R ={(a,b): a ≤ b²} is neither reflexible nor symmetric nor transitive .
8) In the set of all triangles in a plane, show the relation of similarity is an equivalence relation.
9) Is the relation "is the square of" for the set of natural numbers N an equivalence relation ?
10) Consider the following properties of relations : symmetric (S), Transitive (T), Reflexive (R) and Equivalence (E).
A relation may may have all, some or none of these properties. For each part, state whether the relation has some or all the properties mentioned by writing the letters S, T, R, E in the space provided. Write N, if none of the properties satisfy .
a) "is smaller than"
b) " is the father of".
c) " is parallel to" for set of straight lines.
d) " is a multiple of" for a set of positive integers .
e) "is congruent to".
11) Answer true or false :
The relation "is congruent to" in a set of triangles is a transitive relation.
12) if R is a relation in N x N defined by (a,b) R (c,d) if and only of a+ d = b + c, show that R is an Equivalence relation.
13) Prove 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.
14) Let l be the set of all integers and R be the relation on I defined by aRb iff (a + b) is an even integer for all a, b ∈ I. Prove that R is an equivalence relation.
15) Let I be the set of all integers and R be the relation on I defined by R = {(x,y): x, y ∈ I, x - y is divisible by 11}, Prove that R is an Equivalence relation.
FUNCTION
EXERCISE - A
1) Which of the following relations are functions ? Give reasons , In case of a function, determine its domain and range.
i) {(1,-2),(3,7),(4,-6),(8,11)}
ii) {(1,0),(1,-1),(2,3),(4,10)}
iii) {(a,b),(b,c),(c,d),(d,c)}
2) State whether or not each of the following diagrams defines a function of
A={a,b,c} into B={x,y,z}.
3) Which of the following relations are functions?
i) 3x +2.
ii) a is the capital of b ∈ B and B is the set of all countries, a∈ A and A is the set of capital of cities of countries.
iii) y < x +3.
iv) y is the Maths teacher of x where x represents any pupil taking up Maths in a school.
v) y is a Maths pupil of x, where x represents any Maths teacher in a school.
4) State the domain of these functions:
a) f: x ---> 5x
b) g: x ---> 5x, x ∈ Z
c) h: x ---> 2/(x -7).
d) F: x ---> 5x, x ∈ {0,1,2}
e) f: x ---> x/5
f) F: x ---> 6/x.
g) H: x ---> x²+ 5x - 6.
h) g: x ---> (x- 4)/{(x -3)(x +6)}.
i) g: x ---> x/1, x ∈ {2,4,6}
j) g: x ---> 1/x, x ∈ R.
EXERCISE - B
1) A function f given as f: {(2,7,(3,4),(7,9),(-1,6),(0,2),(5,3).
Is this function one-one onto ?
Interchange the order of the elements in the ordered pairs and form the new relation. Is this relation a function ? If it is a function, is it one-one onto.
2) Determine if each function is one-one .
i) To each person on the Earth assign the number which corresponds to his age.
ii) To each country in the world assign the latitude and longitude of its capital.
iii) To each book written by only one author assign the author.
iv) To each country in the world which has a prime minister assign its prime minister.
3) Let f: A--> B. Find f(A), i.e., the range of f, if f is an onto function.
4) Show that the function f: R---> R given by f(x)= cosx for x ∈ R, is neither one-one nor onto.
5) Let A={-1,1}. Let functions f, g and h of A onto A be defined by:
i) f(x)= x.
ii) g(x)= x³
iii) h(x)= sinx.
Which function, if any, is onto?
6) Given A={2,3,4}, B={2,5,6,7}, construct an example of each of the following.
i) A one-one mapping from A to B.
ii) A mapping from A to B which is not one-one .
iii) A mapping from B to A .
7) Are the following ssta of ordered pairs functions ? If so, examine whether the mapping is onto or one-one.
i) {(x,y): x is a person, y is the mother of x}
ii) {(x,y): a is a person, b is an ancestor of a}
8) Is the function f: N --> N (N is the set of natural numbers ) defined by
f(n)= 2n +3 for all n ∈N onto ?
9) let A={x =0 ≤ x ≤2} and B={1}. Give an example of a function from A to B. Can you define a function from B to A which is onto ? Give reasons for your answer.
10) Prove that the function f: R ---> R, f(x)= x²+ x is a many-one into function.
11) 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 but not onto.
12) Show that the function f: R---> R : f(x)= 3- 4x is one-one onto and hence bijective.
EXERCISE - C
1) If the function f: N --> N is defined by f(x)= √x, then find f(25)/{f(16)+ f(1)}.
2) If f(x)= x³/2 - x²/2 + x - 16, find f(1/2).
3) If f(x)= 7x⁴- 2x³- 8x -5, find f(-1).
4) If f(x)= 3x -2 when x ≤ 0
x +1 when x > 0
Find f(-1) and f(0).
5) If f(x)= log{(1- x)/(1+ x)}, show that f(a)+ f(b)= f{(a+ b)/(1+ ab)}.
6) If f(x)= 2x √(1- x²), then show that f{sin(x/2)} = sinx.
7) If f(x)= cos(log x), then show that
f(1/x). f(1/y) - (1/2)[f(x/y) + f(xy)]= 0.
8) If y= f(x)= (5x +3)/(4x -5), then show that f(y)= x.
9) State with reason whether the 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)}
10) Let A={3,5,7,9}, B={9,25,49,81} and f: A--> B be given by f(x)= x². Write f and f⁻¹ as a set of ordered pairs.
11) Consider f: {1,2,3}--> {a,b,c} given by f(1)= a, f(2)= b and f(3)= c. Find the inverse f⁻¹. Show that (f⁻¹)⁻¹= f.
12) Consider f: R---> R given by 4x +3. Show that f is invertible and find the inverse of f.
13) Let S={a,b,c} and T={1,2,3}. Find D⁻¹ of the following function F from S to T, if it exists,
i) F={(a,3),(b,2),(c,1)}
ii) F={a,2),(b,1),(c,1)}
14) f= R --> R is a function defined by f(x)= 10x -7. If g= f⁻¹, then find g(x).
15) If f(x)= (1+ x)/(1- x), then show that f⁻¹(y)= (y- 1)/(y +1).
16) If f(x)=(3x +2)/(5x -3), then show that f⁻¹(x)= f(x).
17) If R---> R defined by f(x)= (3x +5)/2 is an invertible function. Find f⁻¹.
18) Consider f: R⁺---> [4, ∞) given by f(x)= x²+ 4. Show that f is invertible with the inverse of f⁻¹ of f given by f⁻¹(y)= √(y - 4}, where R⁺ is the set of all non-negative real numbers.
19) If f(x)= x⅖+ Kx +1, for all x and if it is an even function, find k.
20) If f(x)= x³- (k -2)x²+ 2x, for all x and if it is an odd function, find k.
21) Is there a function f which is both even and odd ?
22) The function f(x)= log(x + √(x²+1)), is
i) an even function
ii) an odd function
iii) a periodic function
iv) Neither an even nor an odd function.
23) Show that (1/x) log[√{x + √(x²+1)}] is an even function.
EXERCISE - D
1) Find (gof)(3), (fog)(1) and (fof)(0) if
a) f(x)=3x -2, g(x)= x².
b) f(x)= |x +2|, g(x)= - x²
c) f(x)= x²-1, g(x)= √x.
2) If f(x)= x +5 and g(x)= x²-3, find the following:
a) f(g(0))
b) g(f(0))
c) f(g(x)).
d) g(f(x)).
e) f(f(x)).
f) g(g(x)).
g) f(f(-5)).
h) g(g(2)).
3) If u(x)= 4x -5, v(x)= x², and f(x)= 1/x, find
a) u(v(f(x))).
b) u(f(v(x)))
c) f(u(v(x))).
4) Find the inclined values, where
g(t)= t²-1 and f(x)= 1+ x.
a) g(f(0))+ f(g(0)).
b) g(f(2) +3)
5) If f(x)= (2x +1)/(3x -2), then (fof)(2) is equal to
a) 1 b) 2 c) 3 d) 4
6) If f(x)= (1- x)/(1+ x), then f(f(cos 2θ) is equal to
a) cos2θ b) tan2θ c) sec2θ d) cot2θ
7) If f(x)= sin²x and the composite function g(f(x))= |sinx|, then the function g(x) is equal to
a) - √x b) √x c) √(x -1) d) √(x +1)
8) If f: R---> R and g: R---> R are defined respectively as f(x)= x²+ 3x +1 and g(x)= 2x -3, find
a) fog
b) gof
9) If f: R --> R, f(x)= x², g : R---> R, g(x)= cosx. Find
a) fog
b) gof
And show the fog ≠ gof
10) If f(x)= logx, g(x)= x³, then show that f(g(a))+ f(g(b))= 3f(ab).
11) If f(x)= (25 - x⁴)¹⁾⁴ for 0< x < √5, then show f[f(1/2)]= 2⁻¹·
