Problem 13 - Algebra and Trigonometry

Question

If \(n(A \cup B)=50, n(A \cap B)=10,\) and \(n(B)=20\) find \(n(A) .\)

Step-by-Step Solution

Answer

n(A) = 40

Step 1 - Understand Given Data

Identify the values provided in the problem: - The number of elements in the union of sets A and B: \( n(A \cup B) = 50 \). - The number of elements in the intersection of sets A and B: \( n(A \cap B) = 10 \). - The number of elements in set B: \( n(B) = 20 \).

Step 2 - Apply the Principle of Inclusion-Exclusion

According to the principle of inclusion-exclusion for two sets, the number of elements in the union of sets A and B can be found by the formula:\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]Substitute the given values into the formula.

Step 3 - Substitute Known Values

Substitute the values into the formula from Step 2:\[ 50 = n(A) + 20 - 10 \]

Step 4 - Solve for \(n(A)\)

Rearrange the equation to solve for \( n(A) \):\[ 50 = n(A) + 10 \]Subtract 10 from both sides:\[ n(A) = 50 - 10 \]Calculate the result:\[ n(A) = 40 \]

Key Concepts

set theory

Set theory is a branch of mathematical logic that studies collections of objects, called sets. Sets are used to group objects in a structured way, making it easier to analyze and solve problems involving multiple groups.
In set theory, we typically denote sets with capital letters like A, B, C, etc. Elements within sets are the objects themselves, and are often represented using braces, for example, \(\{ a, b, c \}\). Understanding the relationships between sets, such as union and intersection, is important to solve problems like the one in the exercise.

union and intersection of sets

The concepts of union and intersection are fundamental in set theory.

- **Union of Sets (A \cup B)**: The union of sets A and B, written as \(A \cup B\), includes all elements that belong to either A or B, or both. It represents the total group of elements when both sets are combined.
- **Intersection of Sets (A \cap B)**: The intersection of sets A and B, written as \(A \cap B\), includes only the elements that are common to both A and B.

In our exercise, we have \(n(A \cup B) = 50\) and \(n(A \cap B) = 10\). This tells us the number of elements in the combined sets and the shared elements between them, respectively.

algebraic manipulation

Algebraic manipulation involves rearranging and simplifying mathematical expressions and equations to solve for unknown variables.

Following the principle of inclusion-exclusion, we use algebraic manipulation to find \(n(A)\).

The formula for the principle of inclusion-exclusion for two sets is:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
By substituting our known values, we get:

\[ 50 = n(A) + 20 - 10 \]

To isolate \(n(A)\), we rearrange the equation:

\[ 50 = n(A) + 10 \]

Finally, subtract 10 from both sides:

\[ n(A) = 50 - 10 = 40 \]

element count in sets

Element count in sets refers to the number of individual elements contained in a set, often denoted as \(n(A)\) for set A.

In this exercise, we are given:

- The total number of elements in the union of sets A and B: \(n(A \cup B) = 50\)
- The number of elements in the intersection of sets A and B: \(n(A \cap B) = 10\)
- The number of elements in set B: \(n(B) = 20\)

Using these values, we apply the principle of inclusion-exclusion to find the number of elements in set A. The step of substituting the given values into the formula and performing algebraic manipulation helps us solve for \(n(A)\), which is found to be 40.
This method ensures we properly account for overlapping elements without counting them more than once.