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Number of functions into a power set with a given condition

This type of question, asked 2 times: 30 Jan 2023, 30 Jan 2024. All 2 questions below, oldest first.

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  1. Question 1

    30 Jan 2023

    89
    Let S={1,2,3,4,5,6}\mathbf{S = \{ 1,2,3,4,5,6\}}. Then the number of one-one functions f:S→P(S)f:S \rightarrow P(S), where P(S)P(S) denote the power set of SS, such that f(n)⊂f(m)f(n) \subset f(m) where n<mn < m is

  2. Question 2

    30 Jan 2024

    84
    Let A={1,2,3,….7}\mathbf{A = \{ 1,2,3,\ldots.7\}} and let P(a)P(a) denote the power set of A. If the number of functions f:A→P(A)f:A \rightarrow P(A) such that a∈f(a),∀a∈Aa \in f(a),\forall a \in A is mn,mm^{n},m and n∈Nn\in N and mm is least, then m+nm+n is equal to