Case study (4 Marks)

Question 14 Marks

Method to Find the Sets When Cartesian Product is Given For finding these two sets, we write first element of each ordered pair in first set say A and corresponding second element in second set B (say). Number of Elements in Cartesian Product of Two Sets If there are p elements in set A and g elements in set B, then there will be pq elements in A . B i.e. if n(A) = p and n(B) = q, then n(A . B) = pq.
Based on the above two topic, answer the following questions.

If A . B = {(a, 1), (b, 3), (a, 3), (b, 1), (a, 2), (b, 2)}. Then, A and B are:

{1, 3, 2}, {a, b}
{a, b}, {1, 3}
{a, b}, {1, 3, 2}
None of these

If the set A has 3 elements and set B has 4 elements, then the number of elements in A . B is:

A and B are two sets given in such a way that A . B contains 6 elements. If three elements of A . B are (1, 3), (2, 5) and (3, 3), then A, B are:

{1, 2, 3}, {3, 5}
{3, 5,}, {1, 2, 3}
{1, 2}, {3, 5}
{1, 2, 3}, {5}

The remaining elements of A . B in (iii) is:

(5, 1), (3, 2), (3, 5)
(1, 5), (2, 3), (3, 5)
(1, 5), (3, 2), (5, 3)
None of the above

The cartesian product P . P has 16 elements among which are found (a, 1) and (b, 2). Then, the set P is:

{a, b}
{1, 2}
{a, b,1, 2}
{0, b, 1, 2, 4}

Answer

Solution:

Here, first element of each ordered pair of A . B gives the elements of set A and corresponding second element gives the elements of set B.

$\therefore$ A = {a, b} and B = {1, 3, 2}

Note We write each element only one time in set, if it occurs more than one time.

(d) 12

Solution:

Given, n (A) = 3 and n (B) = 4.

$\therefore$ The number of elements in A . B is:

n(A . B) = n(A) . n(B) = 3 . 4 = 12

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

Solution:

It is given that (1, 3), (2, 5) and (3, 3) are in A . B. It follows that 1, 2, 3 are elements of A and 3, 5 are elements of B.

$\therefore$ A = {1, 2, 3} and B = {3, 5}

(b) (1, 5), (2, 3), (3, 5)

Solution:

$\because$ A = {1, 2, 3} and B = {3, 5}

$\therefore$ A = {1, 2, 3} and B = {3, 5}

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

Hence, the remaining elements of (A . B) are (1, 5), (2, 3), (3, 5).

{a, b,1, 2}

Solution:

Given, n(P . P) = 16

⇒ n(P) . n(P) = 16

⇒ n(P) = 4 ....(i)

Now, as $(\text{a},1)\in\text{P}\cdot\text{P}$

$\therefore\text{a}\in\text{P}$ and $1\in\text{P}$

Again, $(\text{b},2)\in\text{P}\cdot\text{P}$

$\therefore\text{b}\in\text{P}$ and $2\in\text{P}$

$\Rightarrow\text{a},\text{b},1,2\in\text{P}$

From Eq. (i), it is clear that P has exactly four elements.

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Question 24 Marks

Ordered Pairs The ordered pair of two elements a and 3 is denoted by (a, b) : a is first element (or first component) and d is second element (or second component). Two ordered pairs are equal if their corresponding elements are equal. ie. (a, b) = (c, d)

⇒ a = c and b = d

Cartesian Product of Two Sets For two non-empty sets A and B, the cartesian product A . B is the set of all ordered pairs of elements from sets Aand B. In symbolic form, it can be written as

$\text{A}\cdot\text{B}=\{(\text{a},\text{b}):\text{a}\in\text{A},\text{b}\in\text{B}\}$

Based on the above topics, answer the following questions.

If (a - 3, 6 + 7) = (3, 7), then the value of aand d are:

6, 0

3, 7

7, 0

3, -7

If (x + 6, y - 2) = (0, 6), then the value of x and y are:

6, 8

-6, -8

-6, 8

6, -8

If (x + 2, 4) = (5, 2x + y), then the value of x and y are:

-3, 2

3, 2

-3, -2

Let A and B be two sets such that A . B consists of 6 elements. If three elements of A . B are (1, 4), (2, 6) and (3, 6), then

(A . B) = (B . A)

$(\text{A}\cdot\text{B})\neq(\text{B}\cdot\text{A})$

A . B = {(1, 4), (1, 6), (2, 4)}

None of the above

If m(A . B) = 45, then n(A) cannot be

Answer

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MATHS Case study (4 Marks)

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