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Chapter 8: Problem 1

How many elements are in \(A_{1} \cup A_{2}\) if there are 12 elements in \(A_{1}, 18\) elements in \(A_{2},\) and $$ \begin{array}{ll}{\text { a) } A_{1} \cap A_{2}=\emptyset ?} & {\text { b) }\left|A_{1} \cap A_{2}\right|=1 ?} \\ {\text { c) }\left|A_{1} \cap A_{2}\right|=6 ?} & {\text { d) } A_{1} \subseteq A_{2} ?}\end{array} $$

Short Answer

Expert verified
a) 30 elements, b) 29 elements, c) 24 elements, d) 18 elements.

Step by step solution

01

Understanding the Problem

We are given two sets, \(A_{1}\) and \(A_{2}\), and we need to find the number of elements in their union \(A_{1} \cup A_{2}\). This will be done under different conditions: when \(A_{1} \cap A_{2} = \emptyset\), \(\left|A_{1} \cap A_{2}\right| = 1\), \(\left|A_{1} \cap A_{2}\right| = 6\), and when \(A_{1} \subseteq A_{2}\).
02

Using the Union Formula

Recall the formula for the number of elements in the union of two sets: \[|A_{1} \cup A_{2}| = |A_{1}| + |A_{2}| - |A_{1} \cap A_{2}|\] We will apply this formula to each condition.
03

Case a) \(A_{1} \cap A_{2} = \emptyset\)

If \(A_{1} \cap A_{2}\) is empty, then \(|A_{1} \cap A_{2}| = 0\). Thus: \[|A_{1} \cup A_{2}| = 12 + 18 - 0 = 30\]
04

Case b) \(\left|A_{1} \cap A_{2}\right| = 1\)

If there is one element in the intersection, then \[|A_{1} \cup A_{2}| = 12 + 18 - 1 = 29\]
05

Case c) \(\left|A_{1} \cap A_{2}\right| = 6\)

If there are six elements in the intersection, then \[|A_{1} \cup A_{2}| = 12 + 18 - 6 = 24\]
06

Case d) \(A_{1} \subseteq A_{2}\)

If \(A_{1}\) is a subset of \(A_{2}\), all elements of \(A_{1}\) are already in \(A_{2}\). Thus, \[|A_{1} \cup A_{2}| = |A_{2}| = 18\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Union of Sets
In set theory, the union of two sets is a fundamental concept that combines all the unique elements from each set. When you hear about the 'union of sets', think of it as merging two groups where every item from both groups is included.

To determine the number of elements in the union of two sets, we use the union formula:

\[|A \cup B| = |A| + |B| - |A \cap B|\]

This formula accounts for elements that may be present in both sets so they are not counted twice. For example, if you have set \(A\) with 12 elements and set \(B\) with 18 elements, and their intersection (elements they share) contains 1 element, the number of elements in their union would be:
\[|A \cup B| = 12 + 18 - 1 = 29\]
When sets do not share any elements (`disjoint sets`), like in case a):
\[|A \cup B| = 12 + 18 - 0 = 30\]
If all elements of one set are within another (`subset`), like in case d):
\[|A \cup B| = |B| = 18\].
Intersection of Sets
The intersection of sets represents the common elements between sets. When discussing the 'intersection of sets', think about what members both groups share.

For example, given sets \(A_1\) with 12 elements and \(A_2\) with 18 elements, if the sets intersect with 6 elements, you use the union formula:

\[|A_1 \cup A_2| = 12 + 18 - 6 = 24\]
This formula makes sure to subtract the shared elements so they are not counted twice.

Understanding intersection is critical, as it helps identify overlaps in data. If there's no intersection (`disjoint sets`), \(A_1 \cap A_2 = \emptyset\), meaning they share no common elements. When you know how many elements two sets share, you can accurately use the union formula to determine their comprehensive union.

Here are few more conditions to consider:
  • Case b): When exactly 1 element is shared:
    \[|A_1 \cup A_2| = 12 + 18 - 1 = 29\]
  • Case c): When 6 elements are shared:
    \[|A_1 \cup A_2| = 12 + 18 - 6 = 24\]
Understanding the intersection helps in avoiding double counting shared elements.
Subset
A subset is a set where all elements are contained within another set. When set \(A_1\) is a subset of set \(A_2\), written as \(A_1 \subseteq A_2\), every element of \(A_1\) is already present in \(A_2\).

Think of it as a smaller group within a larger group. In terms of our example, if \(A_1\) is a subset of \(A_2\), all 12 elements of \(A_1\) are in \(A_2\). Therefore, the union does not bring new elements:

\[|A_1 \cup A_2| = |A_2| = 18\].
This simplifies understanding the union, as we don鈥檛 need the intersection in our calculation.

When learning this concept, visualize subsets as part of a larger set to grasp how many elements are shared without double counting.

Understanding subsets is crucial, as it provides insight into the hierarchical structure of sets. It shows how sets can be integrated, emphasizing the inherent inclusivity of one set within another.

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