Question

Two finite sets have \(\mathrm{m}\) and \(\mathrm{n}\) elements respectively. The total number of subsets of first set is 48 more than the total number of subsets of the second set and \(\mathrm{f}=\mathrm{I}_{\mathrm{N}}\) then \(\mathrm{f}(\mathrm{m}+\mathrm{n})\) is (a) 9 (b) 10 (c) 48 (d) 15

Short answer

The correct answer is f(10). (b) 10

Step-by-step solution

Total number of subsets of a set

The number of subsets of a set with k elements is given by the formula \(2^k\). Thus, if the first set has m elements and the second set has n elements, then the number of subsets of the first set is \(2^m\) and the number of subsets of the second set is \(2^n\).

Understanding the given condition

According to the condition given in the problem, the total number of subsets of the first set is 48 more than the total number of subsets of the second set. This can be written as: \[2^m = 2^n + 48\]

Solving for m and n

In order to find the value of \(f(m+n)\), we need to find possible values of m and n that satisfy the above equation: \[2^m - 2^n = 48\] We can rewrite this equation as follows: \[2^n(2^{m-n} - 1) = 48\] Since 48 can be factored as 3 * 2^4, there are two possible cases to consider: 1. \(2^n = 3, 2^{m-n} - 1 = 2^4\) In this case, \(n = \log_{2}(3)\) which is approximately 1.585. This value of n is not an integer, hence this case is not possible. 2. \(2^n = 2^4, 2^{m-n} - 1 = 3\) In this case, \(n = 4\) and \(2^{m-n} = 4\), which means \(m = 6\).

Finding the value of f(m+n)

Now that we have found the values of m and n, we can find the value of f(m+n). Since m=6 and n=4, we have: \[f(m+n) = f(6+4) = f(10)\] Thus, the correct answer is f(10). The option that matches our result is: (b) 10

Key concepts

Subsets

In mathematics, a subset is defined as a set where every element is also a member of another set. If you have a set with elements, every possible combination of these elements, including the empty set and the full set itself, counts as a subset.
For example, if you have a set \( A \) containing \( m \) elements, the number of subsets you can make from these \( m \) elements is calculated using the formula \( 2^m \). This includes all possible combinations from zero elements (the empty set) to all \( m \) elements.
To illustrate, consider a set with just two elements \( \{a, b\} \). The subsets would be \( \{\}, \{a\}, \{b\}, \{a, b\} \), leading to a total of \( 2^2 = 4 \) subsets. Understanding subsets helps in exploring more advanced topics such as combinations and binomial equations.

• The empty set is always a subset.
• Every set is a subset of itself.

Exponential Growth

Exponential growth refers to an increase where the growth rate of a mathematical function becomes more rapid in proportion to the growing total number.
This is clearly illustrated in the context of subsets, where increasing the number of elements in a set results in a rapid growth of possible subsets. As demonstrated previously, the number of subsets is \( 2^k \). Here, the number grows exponentially with each additional element because the base, 2, is raised to the power of the number of elements \( k \).
If a set grows from 3 to 4 elements, the number of subsets jumps from \( 2^3 = 8 \) to \( 2^4 = 16 \). That's double the number of subsets with just a single new element! This is what makes the concept exciting and applicable in various fields, including computer science, where data grows exponentially as new datasets are considered.

Algebraic Equations

Algebraic equations are mathematical statements that contain variables, arithmetic operators, and are set equal to one another. They are crucial in solving problems that involve unknown values by allowing you to form relationships between known and unknown quantities.
In our exercise, an algebraic equation arises from equating the number of subsets of two sets: \( 2^m \) and \( 2^n + 48 \). By manipulation, we form the equation \( 2^m = 2^n + 48 \), which is the starting point for solving the given problem.
To solve, you rearrange and factor this equation to find values for \( m \) and \( n \) that satisfy the condition. This involves identifying values that uniquely satisfy mathematical operations, such as the power difference \( 2^{m-n} - 1 \), leading to the specific solutions necessary for the answer. Understanding how to manipulate such equations is fundamental to tackling a wide range of mathematical challenges.