MA.912.LT.5.6

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Prove set relations, including DeMorgan’s Laws and equivalence relations.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Logic and Discrete Theory
Standard: Apply properties from Set Theory to solve problems.
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

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Purpose and Instructional Strategies

In Math for College Liberal Arts, students begin to prove set relations, starting with DeMorgan’s Laws and equivalence relations. In other classes, students will build on this knowledge to include proving other set relations. 
  • Sets can be described in three ways. 
    • Word Description: W is the set of days of the week. 
    • Roster Form: elements are listed in { }. The order of the elements does not matter. W={Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} 
    • Set Builder Notation: {x|x is _______} 
      This is read “the set of all x such that x is _______” 
      W={x|x is a day of the week} 
  • Instruction includes students brainstorming different sets that have the same number of elements in order to create equivalent relations. 
    • For example, students can list two sets with 4 elements each. 
      Schools = {Elementary, Middle, K-8 High} 
      Class = {Freshman, Sophomore, Junior, Senior} 
  • Instruction includes proving equivalence relations by showing one-to-one correspondence 
    • For example, {a, e, i, o, u} is equivalent to {1,2,3,4,5}. 

  • Instruction includes proving DeMorgan’s Laws by drawing Venn Diagrams: 
    (A ∪ B)′ = A′ ∩ B′ 

    Blue represents the complement of B. 
    Yellow represents the complement of A. 
    The Crosshatch represents the intersections of those complements.
    (A ∩ B)′ = A′ ∪ B′

    Blue represents the complement of B. 
    Yellow represents the complement of A. 
    Everything shaded represents the union of those complements.

Common Misconceptions or Errors

  • Students may incorrectly try to distribute the negations into (A ∪ B)′ instead of changing the operator from union to intersection. 
  • Students may incorrectly try to distribute the negations into (A ∩ B)′ instead of changing the operator from intersection to union.

Instructional Tasks

Instructional Task 1 (MTR.5.1) 
  • Use Venn Diagrams to prove DeMorgan’s Laws. 

Instructional Task 2 (MTR.2.1,MTR.4.1)
  • Part A. Write two equivalent sets in word description. 
  • Part B. Exchange sets with a partner, write the roster forms for each set and show the sets are equivalent.

Instructional Items

Instructional Item 1 
  • Show that the sets {green, blue, yellow, red} and {square, triangle, circle, hexagon} are equivalent.

*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))
1207350: Mathematics for College Liberal Arts (Specifically in versions: 2022 - 2024, 2024 and beyond (current))
1212300: Discrete Mathematics Honors (Specifically in versions: 2022 - 2024, 2024 and beyond (current))

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