Let A = {1, 2, 3, 4, 5, 6, 7}. Define B = B={T⊆A:either 1 ∉ T or 2 ∈T} and C={T⊆A:T the sum of all the elements of T is a prime number}. Then the number of elements in the set B∪C is

? ( B ∪ C ) ' = B ' ∪ C ' B' is a set containing sub sets of A containing element 1 and not containing 2. And C is a set containing subsets of A whose sum of elements is not prime. so, we need to calculate number of subsets of {3, 4, 5, 6, 7} whose sum of elements plus 1 is composite. Number of such 5 elements subset = 1 Number of such 4 elements subset = 3 (except selecting 3 or 7) Number of such 3 elements subset = 6 (except selecting {3, 4, 5}, {3, 6, 7}, {4, 5, 7} or {5, 6, 7}) Number of such 2 elements subset = 7 (except selecting {3, 7}, {4, 6}, {5, 7}) Number of such 1 elements subset = 3 (except selecting {3, 7}, (4, 6}, {5, 7}) Number of such 0 elements subset = 1 n ( B ∩ C ) = 2 1 ⇒ n ( B ∪ C ) = 2 7 − 2 1 = 1 0 7

Let A = {1, 2, 3, 4, 5, 6, 7}. Define B = $B = {T \subseteq A : e i t h e r 1 \notin T o r 2 \in T}$ and $C = {T \subseteq A : T t h e s u m o f a l l t h e e l e m e n t s o f T i s a p r i m e n u m b e r} .$ Then the number of elements in the set $B \cup C$ is

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Payal Gupta | Contributor-Level 10

8 months ago

$? (B \cup C)' = B' \cup C'$

B' is a set containing sub sets of A containing element 1 and not containing 2.

And C is a set containing subsets of A whose sum of elements is not prime.

so, we need to calculate number of subsets of

{3, 4, 5, 6, 7} whose sum of elements plus 1 is composite.

Number of such 5 elements subset = 1

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