If (  + 1, y –  ) = ( , ), find the values of x and y.

 

If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A × B)?

If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.

State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.

(1) If P = {m, n} and Q = {n, m}, then P × Q = {(m, n), (n, m)}.

(2) If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (x, y) such that x ∈ A and y ∈ B.

(3) If A = {1, 2}, B = {3, 4}, then A × (B ∩ φ) = φ.

If A = {–1, 1}, find A × A × A.

If A × B = {(a, x), (a, y), (b, x), (b, y)}. Find A and B.

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that

(1) A × (B ∩ C) = (A × B) ∩ (A × C).             

(2) A × C is a subset of B × D.

Let A = {1, 2} and B = {3, 4}. Write A × B. How many subsets will A × B have? List them.

Let A and B be two sets such that n(A) = 3 and n(B) = 2.

If (x, 1), (y, 2), (z, 1) are in A × B, find A and B, where x, y and z are distinct elements.

The Cartesian product A × A has 9 elements among which are found (–1, 0) and (0, 1).

Find the set A and the remaining elements of A × A.

Let A = {1, 2, 3...,14}. Define a relation R from A to A by R = {(x, y): 3x – y = 0, where x, y ∈ A}.

Write down its domain, co-domain and range.

Define a relation R on the set N of natural numbers by R = {(x, y): y = x + 5, x is a natural number less than 4; x, y ∈N}.

Depict this relationship using roster form. Write down the domain and the range.

A = {1, 2, 3, 5} and B = {4, 6, 9}.

Define a relation R from A to B by R = {(x, y): the difference between x and y is odd; x ∈ A, y ∈ B}.

Write R in roster form.

The Fig. 2.7 shows a relationship between the sets P and Q.

Write this relation

(1) in set-builder form

(2) roster form.

What is its domain and range?

 

 

 

 

Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by

{(a, b): a, b ∈A, b is exactly divisible by a}.

(1) Write R in roster form

(2) Find the domain of R

(3) Find the range of R.

Determine the domain and range of the relation R defined by

R = {(x, x + 5): x ∈ {0, 1, 2, 3, 4, 5}}.

Write the relation R = {(x, x³): x is a prime number less than 10} in roster form.

Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.

 

Let R be the relation on Z defined by R = {(a, b): a, b ∈ Z, a – b is an integer}. Find the domain and range of R.

1. Which of the following relations are functions?

Give reasons. If it is a function, determine its domain and range.

(1) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

(2) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

(3) {(1, 3), (1, 5), (2, 5)}.

Find the domain and range of the following real functions:

(1) f(x) = -|x|                 

(2) f(x) = √ (9 – x²)

A function f is defined by f(x) = 2x –5. Write down the values of

(1) f (0),               

(2) f (7),                   

(3) f (-3).

The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by

t(C) =  + 32. Find

(1) t (0)           

(2) t (28)        

(3) t (–10)                 

(4) The value of C, when t(C) = 212.

Find the range of each of the following functions.

(1) f(x) = 2 – 3x, x ∈ R, x > 0.

(2) f(x) = x² + 2, x is a real number.

(3) f (x) = x, x is a real number.

 

The relation f is defined by

f(x) =     

{x², 0 ≤ x ≤ 3

{3x, 3 ≤ x ≤ 10

The relation g is defined by

g(x) =    

{x², 0 ≤ x ≤ 2

{3x, 2 ≤ x ≤ 10

Show that f is a function and g is not a function.

If f(x) = x², find. {f (1.1) – f (1)} ÷ (1.1 - 1).

Find the domain of the function f(x) = (x² + 2x + 1) ÷ (x² – 8x + 12)

Find the domain and the range of the real function f defined by

f(x) = √ (x − 1)

Find the domain and the range of the real function f defined by

f (x) = |x – 1|

Let f = {(x, x² ÷ (1 + x²)): x ∈ R} be a function from R into R. Determine the range of f.

Let f, g: R → R be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Find f + g, f – g and .

Let f = {(1, 1), (2, 3), (0, –1), (–1, –3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b.

Determine a, b.

Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b²}. Are the following true?

(1) (a, a) ∈ R, for all a ∈ N

(2) (a, b) ∈ R, implies (b, a) ∈ R

(3) (a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R.

Justify your answer in each case.

Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}.

Are the following true?

(1) f is a relation from A to B

(2) f is a function from A to B.

Justify your answer in each case.

Let f be the subset of Z × Z defined by f = {(ab, a + b): a, b ∈ Z}. Is f a function from Z to Z: justify your answer.

Let A = {9, 10, 11, 12, 13} and let f: A → N be defined by f(n) = the highest prime factor of n.

Find the range of f.