Explore relationships and patterns and make arguments about relationships between sets using Venn Diagrams.

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}.

- For example, the Venn Diagram below shows that the shaded region represents
the complement of

- 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*)′.

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

This benchmark is part of these courses.

7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current))

1210305: Mathematics for College Statistics (Specifically in versions: 2022 and beyond (current))

1207350: Mathematics for College Liberal Arts (Specifically in versions: 2022 and beyond (current))

1212300: Discrete Mathematics Honors (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

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## Lesson Plans

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