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Number of subsets of a set satisfying a condition

This type of question, asked 2 times: 25 Jul 2022, 26 Jul 2022. All 2 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    25 Jul 2022

    171
    Let A={1,2,3,4,5,6,7}A=\{1,2,3,4,5,6,7\}. Define B={T⊆A:B=\{T\subseteq A: either 1∉T1\notin T or 2∈T}2\in T\} and C={T⊆A:C=\{T\subseteq A: the sum of all the elements of TT is a prime number}\}. Then the number of elements in the set B∪CB \cup C is

  2. Question 2

    26 Jul 2022

    171
    Let A={1,2,3,4,5,6,7}\mathbf{A = \{ 1,2,3,4,5,6,7\}} and B={3,6,7,9}\mathbf{B = \{ 3,6,7,9\}}. Then the number of elements in the set {C⊆A:C∩B≠ϕ}\mathbf{\{ C \subseteq A:C \cap B \neq \phi\}} is