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Relations and Functions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsRelations and Functions4 Marks
Question
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.
  1. If $A . B = \{(a, 1), (b, 3), (a, 3), (b, 1), (a, 2), (b, 2)\}.$ Then, $A$ and $B$ are:
  1. $\{1, 3, 2\}, \{a, b\}$
  2. $\{a, b\}, \{1, 3\}$
  3. $\{a, b\}, \{1, 3, 2\}$
  4. None of these
  1. If the set $A$ has $3$ elements and set $B$ has $4$ elements, then the number of elements in $A . B$ is:
  1. $3$
  2. $4$
  3. $7$
  4. $12$
  1. $A$ 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. $\{1, 2, 3\}, \{3, 5\}$
  2. $\{3, 5,\}, \{1, 2, 3\}$
  3. $\{1, 2\}, \{3, 5\}$
  4. $\{1, 2, 3\}, \{5\}$
  1. The remaining elements of $A . B$ in $(iii)$ is:
  1. $(5, 1), (3, 2), (3, 5)$
  2. $(1, 5), (2, 3), (3, 5)$
  3. $(1, 5), (3, 2), (5, 3)$
  4. None of the above
  1. The cartesian product $P . P$ has $16$ elements among which are found $(a, 1)$ and $(b, 2).$ Then, the set $P$ is:
  1. $\{a, b\}$
  2. $\{1, 2\}$
  3. $\{a, b,1, 2\}$
  4. $\{0, b, 1, 2, 4\}$
✓

Answer

$(c)\ \{a, b\}, \{1, 3, 2\}$
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.
  1. $(d) 12$
Solution:
Given, $n (A) = 3$ and $n(B) = 4.$
$\therefore$ The number of elements in $A . B$ is:
$n(A . B) = n(A) . n(B) = 3 . 4 = 12$
  1. $(a) \{1, 2, 3\}, \{3, 5\}$
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 $B = \{3, 5\}$
  1. $(b) (1, 5), (2, 3), (3, 5)$
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 $(1, 5), (2, 3), (3, 5).$​​​​​​​
  1. $\{a, b,1, 2\}$
Solution:
Given, $n(P . P) = 16$
$\Rightarrow n(P) . n(P) = 16$
$\Rightarrow n(P) = 4 ....(i)$
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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