General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Logic and Discrete Theory
Date Adopted or Revised: 08/20
Status: State Board Approved
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In Math for College Liberal Arts students explore relationships between two sets using Venn
Diagrams. In other classes students will extend this exploration to include relationships between
three or more sets.
- Instruction includes usage of Venn Diagrams to represent relationships between sets. The universal set U is represented by a rectangle and the sets within the universe are represented by circles.
- In a Venn Diagram, the complement, A′, is represented by the shaded area.
- For example, U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, A = {0, 1, 3, 4, 5, 7, 9} and B = {0, 1, 2, 4, 6, 8, 10}, then A′ = {2, 6, 8, 10, 11, 12, 13}. The Venn Diagram shows the shaded region for A′.
- In a Venn Diagram, the union of sets A and B, A ∪ B, is represented by the shaded area.
- For example, A = {0, 1, 3, 4, 5, 7, 9, 11} and B = {0, 1, 2, 4, 6, 8, 10, 12}, then A ∪ B is {0, 1, 3, 4, 5, 7, 9, 11} ∪ {0, 1, 2, 4, 6, 8, 10, 12} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. The Venn Diagram of this is the following where the shaded region represents A ∪ B:
- In a Venn Diagram, the intersection of sets A and B, A ∩ B, is represented by the shaded area.
- For example, A = {0, 1, 3, 4, 5, 7, 9, 11} and B = {0, 1, 2, 4, 6, 8, 10, 12}, then A ∩ B is {0, 1, 3, 4, 5, 7, 9, 11} ∩ {0, 1, 2, 4, 6, 8, 10, 12} = {0, 1, 4}. The Venn Diagram of this is the following where the shaded region represents A ∩ B:
- Operations can be combined, following the order of operations.
- For example, the Venn Diagram below shows that the shaded region represents the complement of A ∪ B. A = {0, 1, 3, 4, 5, 7, 9} and B = {0, 1, 2, 4, 6, 8, 10, 12}, then A ∪ B is {0, 1, 3, 4, 5, 7, 9} ∪ {0, 1, 2, 4, 6, 8, 10, 12} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,12}. The (A ∪ B) ′ = {11, 13, 14}.
- Instruction includes finding the difference of two sets. Order matters when finding the difference of two sets.
- In a Venn Diagram, A − B is represented by the shaded area.
- In a Venn Diagram, B − A is represented by the shaded area.
Common Misconceptions or Errors
- Students may repeat elements that are in both sets when writing the union.
- Students may confuse union and intersection.
- Students may incorrectly apply the word “and” when applying set operations.
Instructional Tasks
Instructional Task 1 (MTR.4.1)- Find the following sets using the given Venn Diagram.
Part A. A
Part B. A′
Part C. B
Part D. B′
Part E. A ∪ B
Part F. A ∩ B
Part G. (A ∩ B)′
Part H. A − B
Part I. Describe a set of operations that would result in the set {w, y}
Part J. Describe a set of operations that would result in the set {e, k, m, r, t}
Part K. Describe a set of operations that would result in the set
{a, b, c, d, f, g, h, j, n, p, w, y}
Instructional Items
Instructional Item 1- Find (A ∪ B)′.
