Question 1

How many elements will be there in the Cartesian Product of A and B, if number of elements in A and B are respectively 10 and 7?

Accepted Answer

70

Question 2

Range of greatest integer function given by f(x) = [x] is

Accepted Answer

I or Z

Question 3

If f(x) = x 2 , defined on N, then functionf(x) is of the type

Accepted Answer

ONE-ONE, ONTO

Question 4

Let relation R is defined as R = { ( x, x+2 ) /x belongs to A }, where A= {0,1,2,3}. Then range of the relation is

Accepted Answer

{2,3,4,5}

Question 5

R = { ( x , y )/ x+2y =5 } is defined on set A = { 1, 2, 3, 4, 5 } then the domain will be

Accepted Answer

{1,2,3}

Question 6

If $\left(\frac{x}{3} + 1, y - \frac{2}{3}\right) = \left(\frac{5}{3}, \frac{1}{3}\right)$, find the values of $x$ and $y$.

Accepted Answer

Given $\left(\frac{x}{3} + 1, y - \frac{2}{3}\right) = \left(\frac{5}{3}, \frac{1}{3}\right)$, equate components:

$\frac{x}{3} + 1 = \frac{5}{3}$ and $y - \frac{2}{3} = \frac{1}{3}$.

From first equation:
$\frac{x}{3} = \frac{5}{3} - 1 = \frac{5}{3} - \frac{3}{3} = \frac{2}{3}$
$\Rightarrow x = 2$.

From second equation:
$y = \frac{1}{3} + \frac{2}{3} = 1$.

Hence, $x = 2$ and $y = 1$. — Equate the corresponding components of the ordered pairs and solve the resulting equations step-by-step.

Question 7

If the set A has 3 elements and the set $\mathrm{B} = \{3, 4, 5\}$, then find the number of elements in $(\mathrm{A} \times \mathrm{B})$.

Accepted Answer

Number of elements in $\mathrm{A}$ is 3.
Number of elements in $\mathrm{B}$ is 3.

Number of elements in $\mathrm{A} \times \mathrm{B} = n(\mathrm{A}) \times n(\mathrm{B}) = 3 \times 3 = 9.$ — The Cartesian product of two sets has number of elements equal to the product of the number of elements in each set.

Question 8

If $\mathrm{G} = \{7, 8\}$ and $\mathrm{H} = \{5, 4, 2\}$, find $\mathrm{G} \times \mathrm{H}$ and $\mathrm{H} \times \mathrm{G}$.

Accepted Answer

Given $\mathrm{G} = \{7, 8\}$ and $\mathrm{H} = \{5, 4, 2\}$.

$\mathrm{G} \times \mathrm{H} = \{(7,5), (7,4), (7,2), (8,5), (8,4), (8,2)\}$.

$\mathrm{H} \times \mathrm{G} = \{(5,7), (5,8), (4,7), (4,8), (2,7), (2,8)\}$. — Cartesian product $A \times B$ is the set of all ordered pairs $(a,b)$ with $a \in A$ and $b \in B$.

Question 9

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

(i) If $\mathrm{P} = \{m, n\}$ and $\mathrm{Q} = \{n, m\}$, then $\mathrm{P} \times \mathrm{Q} = \{(m, n), (n, m)\}$.

(ii) If A and B are non-empty sets, then $\mathrm{A} \times \mathrm{B}$ is a non-empty set of ordered pairs $(x, y)$ such that $x \in \mathrm{A}$ and $y \in \mathrm{B}$.

(iii) If $\mathrm{A} = \{1, 2\}$, $\mathrm{B} = \{3, 4\}$, then $\mathrm{A} \times (\mathrm{B} \cap \phi) = \phi$.

Accepted Answer

(i) False.

Correct statement: $\mathrm{P} \times \mathrm{Q} = \{(m,m), (m,n), (n,m), (n,n)\}$ because Cartesian product contains all ordered pairs with first element from $\mathrm{P}$ and second from $\mathrm{Q}$.

(ii) True.

(iii) True. Since $\mathrm{B} \cap \phi = \phi$, $\mathrm{A} \times \phi = \phi$. — For (i), Cartesian product includes all pairs from elements of $\mathrm{P}$ and $\mathrm{Q}$, not just two pairs.

For (ii), definition of Cartesian product.

For (iii), intersection with empty set is empty, so product is empty.

Question 10

If $\mathrm{A} = \{-1, 1\}$, find $\mathrm{A} \times \mathrm{A} \times \mathrm{A}$.

Accepted Answer

Given $\mathrm{A} = \{-1, 1\}$.

The Cartesian product $\mathrm{A} \times \mathrm{A} \times \mathrm{A}$ consists of all ordered triples $(a,b,c)$ where each of $a,b,c \in \mathrm{A}$.

Number of elements = $2 \times 2 \times 2 = 8$.

Elements are:
$(-1,-1,-1), (-1,-1,1), (-1,1,-1), (-1,1,1), (1,-1,-1), (1,-1,1), (1,1,-1), (1,1,1)$. — Cartesian product of three sets is the set of all ordered triples with elements from each set respectively.

Question 11

If $\mathrm{A} \times \mathrm{B} = \{(a, x), (a, y), (b, x), (b, y)\}$. Find A and B.

Accepted Answer

Given $\mathrm{A} \times \mathrm{B} = \{(a, x), (a, y), (b, x), (b, y)\}$.

From the pairs, elements of $\mathrm{A}$ are $\{a, b\}$ and elements of $\mathrm{B}$ are $\{x, y\}$.

Hence, $\mathrm{A} = \{a, b\}$ and $\mathrm{B} = \{x, y\}$. — The first elements of ordered pairs form set $\mathrm{A}$, second elements form set $\mathrm{B}$.

Question 12

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

(i) $\mathrm{A} \times (\mathrm{B} \cap \mathrm{C}) = (\mathrm{A} \times \mathrm{B}) \cap (\mathrm{A} \times \mathrm{C})$.

(ii) $\mathrm{A} \times \mathrm{C}$ is a subset of $\mathrm{B} \times \mathrm{D}$.

Accepted Answer

(i) First, find $\mathrm{B} \cap \mathrm{C}$:

$\mathrm{B} = \{1, 2, 3, 4\}$, $\mathrm{C} = \{5, 6\}$.

Since no common elements, $\mathrm{B} \cap \mathrm{C} = \emptyset$.

Therefore, $\mathrm{A} \times (\mathrm{B} \cap \mathrm{C}) = \mathrm{A} \times \emptyset = \emptyset$.

Now, $\mathrm{A} \times \mathrm{B} = \{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4)\}$.

$\mathrm{A} \times \mathrm{C} = \{(1,5),(1,6),(2,5),(2,6)\}$.

Their intersection $(\mathrm{A} \times \mathrm{B}) \cap (\mathrm{A} \times \mathrm{C}) = \emptyset$ because no common ordered pairs.

Hence, $\mathrm{A} \times (\mathrm{B} \cap \mathrm{C}) = (\mathrm{A} \times \mathrm{B}) \cap (\mathrm{A} \times \mathrm{C}) = \emptyset$.

(ii) $\mathrm{A} \times \mathrm{C} = \{(1,5),(1,6),(2,5),(2,6)\}$.

$\mathrm{B} \times \mathrm{D} = \{(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8)\}$.

All elements of $\mathrm{A} \times \mathrm{C}$ are in $\mathrm{B} \times \mathrm{D}$.

Therefore, $\mathrm{A} \times \mathrm{C} \subseteq \mathrm{B} \times \mathrm{D}$. — Use set operations and definitions of Cartesian products to verify the equalities and subset relation.

Question 13

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

Accepted Answer

First, $\mathrm{A} \times \mathrm{B} = \{(1,3), (1,4), (2,3), (2,4)\}$.

Number of elements in $\mathrm{A} \times \mathrm{B} = 4$.

Number of subsets of a set with 4 elements = $2^4 = 16$.

The subsets are:
1) $\emptyset$
2) \{(1,3)\}
3) \{(1,4)\}
4) \{(2,3)\}
5) \{(2,4)\}
6) \{(1,3),(1,4)\}
7) \{(1,3),(2,3)\}
8) \{(1,3),(2,4)\}
9) \{(1,4),(2,3)\}
10) \{(1,4),(2,4)\}
11) \{(2,3),(2,4)\}
12) \{(1,3),(1,4),(2,3)\}
13) \{(1,3),(1,4),(2,4)\}
14) \{(1,3),(2,3),(2,4)\}
15) \{(1,4),(2,3),(2,4)\}
16) \{(1,3),(1,4),(2,3),(2,4)\}. — Number of subsets of a set with n elements is $2^n$. List all subsets systematically.

Question 14

Let A and B be two sets such that $n(\mathrm{A}) = 3$ and $n(\mathrm{B}) = 2$. If $(x, 1), (y, 2), (z, 1)$ are in $\mathrm{A} \times \mathrm{B}$, find A and B, where $x, y$ and $z$ are distinct elements.

Accepted Answer

Given $n(\mathrm{A}) = 3$, $n(\mathrm{B}) = 2$.

Since $(x,1), (y,2), (z,1) \in \mathrm{A} \times \mathrm{B}$, the second elements 1 and 2 belong to $\mathrm{B}$.

So, $\mathrm{B} = \{1, 2\}$.

The first elements $x, y, z$ are distinct elements of $\mathrm{A}$.

Since $n(\mathrm{A})=3$, $\mathrm{A} = \{x, y, z\}$ with distinct elements. — Use the definition of Cartesian product and given elements to identify sets.

Question 15

The Cartesian product  $\mathrm{A} \times \mathrm{A}$  has 9 elements among which are found  $(-1, 0)$  and  $(0, 1)$ . Find the set  $\mathrm{A}$  and the remaining elements of  $\mathrm{A} \times \mathrm{A}$ .

Accepted Answer

Given $\mathrm{A} \times \mathrm{A}$ has 9 elements.

Since $n(\mathrm{A} \times \mathrm{A}) = n(\mathrm{A})^2 = 9$, so $n(\mathrm{A}) = 3$.

Given $(-1,0)$ and $(0,1)$ are in $\mathrm{A} \times \mathrm{A}$, so $-1, 0, 1 \in \mathrm{A}$.

Hence, $\mathrm{A} = \{-1, 0, 1\}$.

Remaining elements of $\mathrm{A} \times \mathrm{A}$ are all ordered pairs with elements from $\{-1,0,1\}$ except $(-1,0)$ and $(0,1)$:

$(-1,-1), (-1,1), (0,-1), (0,0), (1,-1), (1,0), (1,1)$. — Use the fact that the number of elements in $\mathrm{A} \times \mathrm{A}$ is square of number of elements in $\mathrm{A}$. Use given pairs to find elements of $\mathrm{A}$.

Relations and Functions

Relations and Functions — Study Notes

2.1 Introduction

2.2 Cartesian Products of Sets

Remarks on Cartesian Products and Examples

Practice Questions — Relations and Functions

All 14 Chapters in Mathematics