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Let the set = {2,3,4,..., 25}. For each k $$\epsilon$$ P, define Q(k)= {x ∈ P such that x > k and k divides x}. Then the number of elements in the set $$ P - U_{k = 2}^{25}$$ Q(K) is
Correct Answer: 9
$$P=\left\{2,3,4,...,25\right\}$$
Q(k)= {x ∈ P such that x > k and k divides x}. Now, we will have multiple values of k.
Case 1: When k = 2, Q(2) = {4,6,8,10,....,24}
Case 2: When k = 3, Q(3) = {6,9,12,15,....,24}
Case 3: When k = 4, Q(4) = {8,12,16,....24}
Case 4: When k = 5, Q(5) = {10,15,20,25}
Now, in the above 4 cases, we can see that - 2, 3, and 5 are not included, and in the next values of k, since each value of Q(k) is greater than k, thus 2, 3 and 5 will never be included. Now, in further values of k, the numbers which are prime will never be included in any set, because in Q(k), each element is such that x > k, and k divides x. But prime number is divisible by only 1 and itself. And the prime number cannot include itself as well, because x > k. Thus, $$U_{k=2}^{25}Q\left(k\right)$$ will contains all the numbers from 2 to 24, which are not prime. Thus, -
$$U_{k=2}^{25}Q\left(k\right)=Q\left(2\right)∪Q\left(3\right)∪Q\left(4\right)∪.....∪Q\left(25\right)$$
$$U_{k=2}^{25}Q\left(k\right)=\left\{4,6,8,9,10,12,14,15,16,18,20,21,22,24,25\right\}$$
$$P-U_{k=2}^{25}Q\left(k\right)=\left\{2,3,5,7,11,13,17,19,23\right\}$$
Thus, 9 elements are there in the above set.
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