4.2 Union of Sets
The union of sets combines all elements from two or more given sets into a single set. Written using the symbol \(\cup\), the union \(P \cup Q\) represents all the elements that belong to set \(P\), set \(Q\), or both sets \(P\) and \(Q\).
Learning Objectives
- Determine and describe the union of sets using various representations including set notations, listings, and Venn diagrams.
- Determine and describe the complement of the union of sets.
- Solve real-world mathematical problems involving the union of sets and the complement of the union of sets.
Shading Regions for Union and Complement of Union in Venn Diagrams
Compare the shaded region for \(A \cup B\) with the shaded region for \((A \cup B)'\).
Shade the entire interior of both circles.
Shade the region outside both circles, inside \(\xi\).
Brief Explanation
What is the union of sets?
The union of sets \(P\) and \(Q\), written \(P \cup Q\), represents all the elements in set \(P\) or set \(Q\) or in both sets \(P\) and \(Q\). In set builder notation, \(A \cup B = \{x : x \in A \text{ or } x \in B\}\).
What is the complement of the union?
The complement of the union, written \((A \cup B)'\), refers to all the elements in the universal set \(\xi\) that are not in set \(A\) and not in set \(B\).
| Representation | Description | Example |
|---|---|---|
| Listing elements | Write each element once, even if it appears in more than one set. | \(P = \{\text{factors of } 24\}\), \(Q = \{\text{multiples of } 3 \text{ less than } 20\}\): \(P \cup Q = \{1, 2, 3, 4, 6, 8, 9, 12, 15, 18, 24\}\) |
| Venn diagram | Shade the entire region covered by all the sets involved. | Shade circles \(P\) and \(Q\) completely. |
| Complement of union | Shade the region inside \(\xi\) but outside all the sets. | \((A \cup B)'\) is outside both \(A\) and \(B\). |
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
Given set \(A = \{2, 3, 5, 7\}\) and set \(B = \{1, 3, 5, 7, 9\}\). List all the elements of \(A \cup B\).
Given universal set \(\xi = \{1, 2, 3, 4, 5, 6, 7, 8\}\), \(P = \{2, 4, 6, 8\}\), and \(Q = \{6, 7, 8\}\). Find \((P \cup Q)'\).
In a class of 40 pupils, 32 pupils are either from Sabah or Sarawak, such that \(n(B \cup T) = 32\). How many pupils are not from any of these two states?
Common Mistakes
- Writing duplicate elements repeatedly when listing the union of sets, for example writing \(\{3, 3\}\) instead of \(\{3\}\).
- Confusing the union symbol \(\cup\) with the intersection symbol \(\cap\).
- Forgetting to subtract the intersection when finding “elements belonging to a set only”, which leads to double counting.
- Shading the intersection instead of all involved circles when asked to shade the union on a Venn diagram.
- Confusing \((A \cup B)'\) with \(A' \cup B'\).
Misconceptions
The union of two sets requires an element to belong to both sets simultaneously.
The union includes elements that belong to set \(A\), set \(B\), or both. Being in at least one set is sufficient.
\((A \cup B)'\) means the elements in set \(A\) that are not in set \(B\).
\((A \cup B)'\) represents all elements in the universal set \(\xi\) that are outside both set \(A\) and set \(B\).
\(n(A \cup B) = n(A) + n(B)\) always holds true.
If \(A\) and \(B\) share common elements, adding \(n(A)\) and \(n(B)\) counts the intersection twice. The formula is \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\).
Summary
- The union of sets \(A\) and \(B\) (\(A \cup B\)) is the set containing all elements that belong to \(A\), \(B\), or both.
- The complement of the union of sets \(A\) and \(B\) (\((A \cup B)'\)) consists of all elements in the universal set \(\xi\) that are not in \(A \cup B\).
- On Venn diagrams: \(A \cup B\) shades all regions inside circle \(A\) and circle \(B\); \((A \cup B)'\) shades the region outside both circles \(A\) and \(B\) within \(\xi\).
- For problem-solving, identify the non-overlapping component regions starting from the innermost intersection to determine counts accurately.
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 Given the universal set \(\xi = \{x : x \text{ is an integer, } 1 \le x \le 10\}\), set \(A = \{2, 4, 6, 8\}\), and set \(B = \{3, 6, 9\}\). What is \((A \cup B)'\)?
2 The universal set \(\xi = J \cup K \cup L\) is divided into five regions only. If \(n(J \text{ only}) = 7\), \(n(J \cap K) = 2\), \(n(K \text{ only}) = x + 2\), \(n(K \cap L) = 3\), \(n(L \text{ only}) = x + 5\), and \(n(\xi) = 25\), what is the value of \(x\)?
B. Fill in the Blanks
3 The symbol used to denote the union of sets is .
4 In a survey of 200 customers, 114 buy non-fiction books, 52 buy fiction books, and 27 buy both. The number of customers who buy fiction books only is .
C. Drag and Drop: Match Set Notation with Description
Drag each notation to the correct description. On a phone, tap the answer choice first, then tap the description box.
