Set Operations Practice: Union, Intersection, and Complement

Algebra and Function Practice

Set operations combine or compare collections of distinct items. This practice set focuses on three core operations using sets written in roster form, such as A = {1, 2, 3}. The union A ∪ B contains every element that appears in A or B or both. The intersection A ∩ B contains only elements that appear in both A and B. The complement of A, written A′ when the universal set U is understood, contains every element of U that is not in A. Combined expressions such as (A ∪ B)′ or A ∩ B′ require working from the inside out, applying one operation at a time and tracking the universal set carefully. All sets here use distinct elements, and the universal set is stated explicitly for every complement task. Practice with these operations strengthens logical reasoning and prepares you for functions, logic, and data classification. Each exercise is self-contained: the sets are listed in the prompt, so no external diagram is required. When a task mentions a diagram, a text description of the labeled regions is provided instead.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

Core definitions and notation

A set is a collection of distinct objects, called elements. In roster form, elements are listed inside braces, for example B = {2, 4, 6}. The universal set U is the reference set that contains all elements under consideration for a given problem. The union A ∪ B is the set of elements belonging to A, to B, or to both. The intersection A ∩ B is the set of elements belonging to both A and B. The complement A′ is the set of elements in U that are not in A. The empty set, written ∅, contains no elements.

Order does not matter in a set, and repeated elements are not listed twice. For example, {1, 2} and {2, 1} are the same set. When combining operations, parentheses indicate which operation to perform first. For instance, (A ∪ B)′ means find the union first, then take the complement of that result within U.

1. Identify the universal set U and each given set in roster form.
2. For union, list every element from both sets without repeating any element.
3. For intersection, list only elements that appear in both sets.
4. For complement, list elements of U that are not in the specified set.
5. For combined expressions, evaluate the operation inside parentheses first, then apply the remaining operation.

Working with combined operations

Combined expressions such as A ∩ B′ or (A ∪ B)′ require careful sequencing. A useful strategy is to rewrite the expression in words before computing. For example, A ∩ B′ means elements that are in A and also not in B. Computing B′ first and then intersecting with A often reduces errors. Similarly, (A ∪ B)′ means elements of U that are in neither A nor B.

Keep the universal set visible throughout the problem. A common error is to take a complement relative to the wrong set or to forget that the complement depends on U. After computing an answer, verify that every listed element belongs to U and that no element is repeated.

1. Rewrite the expression in words to clarify the intended operation order.
2. Compute the innermost operation first, listing the resulting set explicitly.
3. Apply the next operation to that result and the remaining set.
4. Check that the final set contains only elements from U and has no duplicates.
5. Compare the result with a second method, such as listing elements by membership conditions.

Checking your work

A reliable check is to test each element of U against the final expression. For A ∪ B, an element should be included exactly when it is in A or in B. For A ∩ B, it should be included exactly when it is in both. For A′, it should be included exactly when it is not in A. This element-by-element test catches missing or extra elements.

Another check is to compare the size of the result with expected bounds. The union cannot have fewer elements than either input set, and the intersection cannot have more elements than either input set. The complement of A within U has size equal to the number of elements in U minus the number of elements in A, provided A is a subset of U.

1. List every element of U and mark whether it satisfies the final condition.
2. Count the elements in the result and compare with the expected size.
3. Verify that no element appears twice and that all elements belong to U.
4. Recompute using a different operation order where possible, such as using A ∩ B′ versus A minus B.

Worked example

Let U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3, 4}, and B = {3, 4, 5}. Find (A ∪ B)′.

Four practice exercises

Common mistakes

• Including duplicate elements when forming a union. List each distinct element once. For example, if A = {1, 2} and B = {2, 3}, then A ∪ B = {1, 2, 3}, not {1, 2, 2, 3}.
• Taking a complement relative to the wrong universal set. Write U at the top of your work and check that every complement uses that exact U. If U changes, the complement changes.
• Evaluating combined operations in the wrong order. Follow parentheses first. Rewrite the expression in words, such as 'in A and not in B' for A ∩ B′, before computing.
• Confusing intersection with union when sets overlap heavily. For intersection, ask whether an element appears in both sets. For union, ask whether it appears in at least one set. Test each element separately.

Self-check rubric

• Correctly identifies the universal set and all given sets in roster form.
• Applies union by listing each distinct element from both sets exactly once.
• Applies intersection by listing only elements common to both sets.
• Applies complement relative to the stated universal set.
• Evaluates combined operations in the correct order, following parentheses.
• Presents final answers as sets in roster form with no duplicates.
• Shows a check, such as testing each element of U against the final condition.

Questions and answers