Let A - x - x is an odd prime number and x -15 and B - x - x is divisible by either 5 or 7- and 3 - x - 15- If P - A - B - Q - A - B and the number of relations from P to Q is 4 k- then the value of k is

A= {x: x is an odd prime number and x 15 } ⇒ A = {3,5,7,11,13} B= { x : x is divisible by either 5 or 7, and 3 ≤ x ≤ 15} ⇒ B= {5,7,10,14,15} P= A ∪ B ={3,5,7 ...

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Byju's Answer

Standard XII

Mathematics

Combination with Restrictions

Let A = x : x...

Question

Let $A = {x : x is an odd prime number and x < 15}$ and $B = {x : x is divisible by either 5 or 7, and 3 \leq x \leq 15}$ . If $P = A \cup B, Q = A \cap B$ and the number of relations from $P$ to $Q$ is $4^{k}$ , then the value of $k$ is

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Solution

$A = {x : x is an odd prime number and x < 15}$
$\Rightarrow A = {3, 5, 7, 11, 13}$
$B = {x : x is divisible by either 5 or 7, and 3 \leq x \leq 15}$
$\Rightarrow B = {5, 7, 10, 14, 15}$

$P = A \cup B = {3, 5, 7, 10, 11, 13, 14, 15}$
$Q = A \cap B = {5, 7}$

Number of relations from $P$ to $Q$
$= 2^{n (P) \cdot n (Q)}$
$= 2^{8 \times 2} = 2^{16} = 4^{8}$
Hence, $k = 8$

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Let A={x:x is an odd prime number and x<15} and B={x:x is divisible by either 5 or 7, and 3≤x≤15}. If P=A∪B,Q=A∩B and the number of relations from P to Q is 4k, then the value of k is

A={x:x is an odd prime number and x<15} ⇒A={3,5,7,11,13} B={x:x is divisible by either 5 or 7, and 3≤x≤15} ⇒B={5,7,10,14,15} P=A∪B={3,5,7,10,11,13,14,15} Q=A∩B={5,7} Number of relations from P to Q =2n(P)⋅n(Q) =28×2=216=48 Hence, k=8

Let $A = {x : x is an odd prime number and x < 15}$ and $B = {x : x is divisible by either 5 or 7, and 3 \leq x \leq 15}$ . If $P = A \cup B, Q = A \cap B$ and the number of relations from $P$ to $Q$ is $4^{k}$ , then the value of $k$ is