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Union of Sets: Definition and Examples | EDU.COM

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Union of Sets: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Union of SetsUnion of Sets: Definition and ExamplesTable of ContentsUnion of Sets Definition of Union of Sets

The union of sets is a fundamental set operation that combines elements from multiple sets. When we take the union of two sets A and B, denoted by A∪BA \cup BA∪B, we create a new set containing all elements that are in set A or set B or in both sets, without any repetition. In simpler terms, the union combines all distinct elements from both sets into a single collection.

The union of sets possesses several important properties that help us understand and work with them. These properties include the commutative property (A∪B=B∪AA \cup B = B \cup AA∪B=B∪A), associative property ((A∪B)∪C=A∪(B∪C)(A \cup B) \cup C = A \cup (B \cup C)(A∪B)∪C=A∪(B∪C)), idempotent property (A∪A=AA \cup A = AA∪A=A), identity law (A∪∅=AA \cup \emptyset = AA∪∅=A), and domination property (A∪U=UA \cup U = UA∪U=U, where U is the universal set). These properties make operations with sets easier to calculate and understand.

Examples of Union of Sets Example 1: Finding the Union of Two Number Sets Problem:

Find A∪BA \cup BA∪B if A={1,2,4,6}A = \{1, 2, 4, 6\}A={1,2,4,6} and B={2,3,6,7,8,9}B = \{2, 3, 6, 7, 8, 9\}B={2,3,6,7,8,9}.

Step-by-step solution:

Step 1, Write down the elements of both sets. Set A={1,2,4,6}A = \{1, 2, 4, 6\}A={1,2,4,6} and Set B={2,3,6,7,8,9}B = \{2, 3, 6, 7, 8, 9\}B={2,3,6,7,8,9}.

Step 2, Identify the common elements between sets A and B. The common elements are 2 and 6.

Step 3, Combine all elements from both sets, listing each element only once. Take all elements from set A and set B without repeating the common elements.

Step 4, Write the union set. A∪B={1,2,3,4,6,7,8,9}A \cup B = \{1, 2, 3, 4, 6, 7, 8, 9\}A∪B={1,2,3,4,6,7,8,9}.

Example 2: Calculating the Number of Elements in a Union Problem:

In a group, 20 people own a white car, 10 people own a black car, and 5 people own both a white and a black car. How many people own a white car or a black car?

Step-by-step solution:

Step 1, Identify what information we have. Let's use W for the set of people who own a white car and B for the set of people who own a black car.

n(W)=20n(W) = 20n(W)=20 (number of people who own a white car) n(B)=10n(B) = 10n(B)=10 (number of people who own a black car) n(W∩B)=5n(W \cap B) = 5n(W∩B)=5 (number of people who own both a white and a black car)

Step 2, Use the formula for finding the number of elements in a union: n(W∪B)=n(W)+n(B)−n(W∩B)n(W \cup B) = n(W) + n(B) - n(W \cap B)n(W∪B)=n(W)+n(B)−n(W∩B)

Step 3, Substitute the values into the formula: n(W∪B)=20+10−5n(W \cup B) = 20 + 10 - 5n(W∪B)=20+10−5

Step 4, Calculate the result: n(W∪B)=30−5=25n(W \cup B) = 30 - 5 = 25n(W∪B)=30−5=25

Step 5, Write the conclusion: 25 people own a white car or a black car.

Example 3: Identifying Union from a Venn Diagram Problem:

What is A∪BA \cup BA∪B in the following venn diagram?

Union of sets

Step-by-step solution:

Step 1, Identify the elements in each set from the Venn diagram. Universal set U={1,2,3,4,5,6}U = \{1, 2, 3, 4, 5, 6\}U={1,2,3,4,5,6} Set A={2,4,6}A = \{2, 4, 6\}A={2,4,6} Set B={3,4,5,6}B = \{3, 4, 5, 6\}B={3,4,5,6}

Step 2, Understand what the union A∪BA \cup BA∪B means. The union includes all elements that are in either set A or set B or in both.

Step 3, Look at the shaded region in the Venn diagram. The entire shaded area shows all elements in A∪BA \cup BA∪B.

Step 4, List the elements in the union. The elements 2, 3, 4, 5, and 6 are in the shaded region. Note that element 1 is in the universal set but not in either A or B.

Step 5, Write the final answer: A∪B={2,3,4,5,6}A \cup B = \{2, 3, 4, 5, 6\}A∪B={2,3,4,5,6}

Comments(1)MMarketerSallyNovember 6, 2025I've used this glossary page to teach set unions. The examples are super clear and really helped my students grasp the concept!

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