Problem 8 - Algebra and Trigonometry

Question

Multiple Choice The Counting Formula states that if \(A\) and \(B\) are finite sets, then \(n(A \cup B)=\) __________. (a) \(n(A)+n(B)\) (b) \(n(A)+n(B)-n(A \cap B)\) (c) \(n(A) \cdot n(B)\) (d) \(n(A)-n(B)\)

Step-by-Step Solution

Answer

Option (b): n(A) + n(B) - n(A ∩ B)

Step 1: Understand the Problem

The problem is asking to identify the correct form of the counting formula for the union of two finite sets, denoted as (A ∪ B).

Step 2: Recall the Counting Formula for Union

The counting formula for the union of two finite sets states that the number of elements in the union of sets A and B is equal to the sum of the number of elements in each set, minus the number of elements that are common to both sets. Mathematically, this is written as: (A ∪ B) = n(A) + n(B) - n(A ∩ B).

Step 3: Match the Formula with Options

Compare the formula (A ∪ B) = n(A) + n(B) - n(A ∩ B) with the given multiple-choice options.

Step 4: Identify the Correct Option

The correct option matches the formula derived in Step 2, which is option (b): (A ∪ B) = n(A) + n(B) - n(A ∩ B).

Key Concepts

Finite Sets

Understanding what finite sets are is crucial in set theory. A 'finite set' is a set that contains a specific and countable number of elements. For instance, the set of vowels in the alphabet {a, e, i, o, u} is finite because you can count them easily and there are only 5 elements.
In mathematics, the notation for the number of elements in a set A is written as 'n(A)'. If we have a set A with elements {1, 2, 3}, then n(A) = 3.
Finite sets make it simpler to perform various operations, such as unions and intersections, because we can count their elements and apply formulas accordingly.

Union of Sets

The 'union' of two sets, denoted as A ∪ B, is a fundamental concept in set theory. The union contains all the elements that are in set A, set B, or both. Think of it as merging two groups together to form a bigger group.
For example, if A = {1, 2, 3} and B = {3, 4, 5}, then the union A ∪ B = {1, 2, 3, 4, 5}. Notice how the element '3' that is common to both sets is only listed once in the union.
The counting formula tells us how to find the number of elements in the union of two finite sets: (A ∪ B) = n(A) + n(B) - n(A ∩ B). This formula means you add the number of elements in both sets, but then subtract the number in the intersection (those elements counted twice).

Intersection of Sets

Next, let's break down the 'intersection' of two sets, denoted as A ∩ B. The intersection includes only those elements that are present in both sets.
For instance, if A = {1, 2, 3} and B = {3, 4, 5}, then the intersection A ∩ B = {3}.
Understanding intersections is vital for using the counting formula correctly because the formula (A ∪ B) = n(A) + n(B) - n(A ∩ B) requires us to know the number of elements common to both sets.
In our example above, since both sets share the element '3', this is why it appears in the intersection, and its count is subtracted in the formula to avoid double-counting.

Set Theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. It's an essential foundation in mathematics because it underpins various concepts like functions, sequences, and probability.
When studying set theory, you'll often encounter various operations like union, intersection, difference, and complement. These operations help you understand the relationships between different sets.
A great aspect of set theory is that it provides a visual way to represent ideas using Venn diagrams, helping to explain concepts like unions and intersections clearly. For practical problems, set theory is useful in data science, computer science, and logic, providing a structured way to think about collections of objects.
The counting formula is just one of the many powerful tools in set theory that helps us solve complex problems involving multiple sets.