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LMS-Week-3.2-Venn-Diagram

Introduction to Venn Diagrams

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  • Unauthorized use, aside from personal learning, is subject to penalties.

Learning Outcomes

  • By the end of the lesson students should be able to:

    • Define a Venn Diagram.

    • Explain its components and purpose.

    • Create a Venn Diagram correctly.

    • Recognize relationships among concepts.

Definition of Venn Diagram

  • A Venn Diagram uses overlapping circles or shapes to illustrate logical relationships between two or more sets of items.

Representing Subsets in Venn Diagrams

  • Universal Set: U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

  • Subset S = {0, 1, 2}, Subset T = {0, 1, 2, 3, 4}

    • S ⊆ U and T ⊆ U (elements of S and T are in U).

    • S ⊆ T (all elements of S are in T).

  • Representation: Smaller circle (S) is enclosed in the larger circle (T).

    • Enclosed in a rectangle representing the universal set U.

Intersection of Two Sets

  • The intersection of two sets A and B, denoted as A ∩ B, comprises elements common to both sets.

    • Represented in a Venn diagram by overlapping regions.

Example of Intersection

  • Let M = {vowels in the alphabet}, L = {letters in ‘algebra’}.

    • M = {a, e, i, o, u}, L = {a, l, g, e, b, r}.

    • Intersection: M ∩ L = {a, e}.

Union of Two Sets

  • The union of A and B, denoted as A U B, is the set containing elements in either A or B or both.

    • Illustrated by the shaded portion of the circles in a Venn Diagram.

Example of Union

  • For sets M and L defined above:

    • M U L = {a, e, i, o, u, l, g, b, r}.

Complement of a Set

  • The complement of a set A, denoted A’, includes elements in the universal set U not contained in A.

    • Corresponds to the shaded region outside set A in a Venn Diagram.

Example of Complement

  • If U = {days of the week}, and B = {days not included on weekdays}:

    • B’ = {Monday, Tuesday, Wednesday, Thursday, Friday}.

Disjoint Sets

  • Two sets are disjoint if they have no elements in common; their intersection is an empty set ({}).

Example of Disjoint Sets

  • Let A = {2, 4, 6, 8}, B = {1, 3, 5, 7, 9}.

    • No elements shared; thus A and B are disjoint.

Sample Problems

Problem 1: Set Operations

  1. For sets A, B, C, solve:

    1. A ∩ (B U C)

    2. (A’ ∩ B) U C

Problem 2: Identifying Regions

  • Given regions represented by Roman numerals, identify items within each region.

Problem 3: City Visit Survey

  • Survey of 100 people regarding visits to Tagaytay and Baguio and calculate individual and combined visits.

Problem 4: Investments Survey

  • Venn diagram analysis of 500 investors in real estate and forex trading and determine subsets.

Problem 5: Fruit Preference Survey

  • Survey of 62 students liking at least one of apples, bananas, or oranges; analyze preferences and exclusives.

Poll Question

  • Assess understanding of the topic with options ranging from none to a lot.