Question

Let \( A = \{1, 2, 3, \ldots, 7\} \) and let \( P(A) \) denote the power set of \( A \). If the number of functions \( f : A \rightarrow P(A) \) such that \( a \in f(a), \, \forall a \in A \) is \( m^n \), \( m \) and \( n \in \mathbb{N} \) and \( m \) is least, then \( m + n \) is equal to \(\_\_\_\_\_\_\_\_\_\).

Answer 1

Correct Answer: 44

The problem requires determining the value of \(m + n\), where \(m^n\) represents the total count of functions \(f: A \to P(A)\) such that for every \(a \in A\), the condition \(a \in f(a)\) holds. The set \(A\) is defined as \(\{1, 2, 3, \ldots, 7\}\), and \(m\) must be the smallest possible natural number.

Concept Used:

The solution employs principles of combinatorial counting. A function \(f: A \to B\) is characterized by assigning a unique element from \(B\) to each element of \(A\). The total number of such functions is given by \(|B|^{|A|}\).

In this specific problem, the domain is \(A\), and the codomain is \(P(A)\), the power set of \(A\). A constraint is placed on the function, limiting the permissible outputs for each input. To ascertain the total number of valid functions, we must count the valid choices for \(f(a)\) for each \(a \in A\) and then multiply these counts, as these choices are independent.

Step-by-Step Solution:

Step 1: Define the domain and codomain.

The domain is \(A = \{1, 2, 3, 4, 5, 6, 7\}\), with its cardinality \(|A| = 7\).

The codomain is \(P(A)\), the set of all subsets of \(A\). The cardinality of the codomain is \(|P(A)| = 2^{|A|} = 2^7\).

A function \(f: A \to P(A)\) maps each element \(a \in A\) to a subset of \(A\), denoted by \(f(a)\).

Step 2: Determine the number of valid function values based on the condition.

The condition \(a \in f(a)\) for all \(a \in A\) dictates that each element \(a\) must be an element of the subset assigned to it by the function.

Consider an arbitrary element \(a_k \in A\). The subset \(f(a_k)\) must be a subset of \(A\) and must include \(a_k\).

Constructing a subset of \(A\) involves deciding for each of the 7 elements whether to include it. For \(f(a_k)\):

• The inclusion of \(a_k\) is mandatory; there is 1 option.
• For the remaining \(|A| - 1 = 7 - 1 = 6\) elements in \(A\), each can either be included or excluded, providing 2 options per element.

Thus, the number of valid subsets for \(a_k\) is \(1 \times 2^6 = 64\).

Step 3: Compute the total number of such functions.

Since the selection of \(f(a)\) for each \(a \in A\) is independent, the total number of functions is the product of the number of choices for each domain element.

With \(|A| = 7\), and each of the 7 elements having \(2^6\) possible images, the total number of functions is:

\[ \text{Total functions} = \underbrace{(2^6) \times (2^6) \times \cdots \times (2^6)}_{7 \text{ times}} = (2^6)^7 \] \[ \text{Total functions} = 2^{6 \times 7} = 2^{42} \]

Step 4: Express the total count as \(m^n\) with the minimum \(m\).

We are given that the total number of functions equals \(m^n\), where \(m, n \in \mathbb{N}\) and \(m\) is minimized. We have:

\[ m^n = 2^{42} \]

To minimize \(m\), we select the smallest possible base greater than 1, which is the prime base of the number. Here, the base is 2.

Setting \(m = 2\) yields:

\[ 2^n = 2^{42} \implies n = 42 \]

The determined values are \(m = 2\) and \(n = 42\).

Final Computation & Result:

Step 5: Calculate \(m + n\).

Using the values \(m = 2\) and \(n = 42\), the sum is:

\[ m + n = 2 + 42 = 44 \]

The value of \(m + n\) is 44.