Venn Diagram
Venn Diagram and Sets
Venn Diagrams
A Venn diagram is an illustration of the relationships between and among sets, groups of objects that share something in common. Usually, Venn diagrams are used to depict set intersections (denoted by an upside-down letter U). This type of diagram is used in scientific and engineering presentations, in theoretical mathematics, in computer applications, and in statistics. (Contributor, 2005)
Venn diagrams help to visually represent the similarities and differences between two concepts. They have long been recognized for their usefulness as educational tools. Since the mid-20th century, Venn diagrams have been used as part of the introductory logic curriculum and in elementary-level educational plans around the world. (Kenton, 2020)
Venn diagrams are, at a basic level, simple pictorial representations of the relationship that exists between two sets of things. However, they can be much more complex. Still, the streamlined purpose of the Venn diagram to illustrate concepts and groups has led to their popularized use in many fields, including statistics, linguistics, logic, education, computer science, and business. (Kenton, 2020)
Sets
Set, In mathematics and logic, any collection of objects (elements), which may be mathematical (e.g., numbers, functions) or not. A set is commonly represented as a list of all its members enclosed in braces. A set with no members is called an empty, or null, set, and is denoted ∅. Because an infinite set cannot be listed, it is usually represented by a formula that generates its elements when applied to the elements of the set of counting numbers. (Britannica, 2014)
+ Union
The union of two sets is a set containing all elements that are in A or in B (possibly both). For example, {1,2}∪{2,3}={1,2,3}. Thus, we can write x∈(A∪B) if and only if (x∈A) or (x∈B). Note that A∪B=B∪A. (Probability Course, 2020)
The intersection of two sets A and B, denoted by A∩B, consists of all elements that are both in A and B. For example, {1,2}∩{2,3}={2}. (Probability Course, 2020)
The difference (subtraction) is defined as follows. The set A−B consists of elements that are in A but not in B. For example if A={1,2,3} and B={3,5}, then A−B={1,2}. (Probability Course, 2020)
The symmetric difference using Venn diagram of two subsets A and B is a sub set of U, denoted by A △ B and is defined by A △ B = (A – B) ∪ (B – A). Let A and B are two sets. The symmetric difference of two sets A and B is the set (A – B) ∪ (B – A) and is denoted by A △ B. Thus, A △ B = (A – B) ∪ (B – A) = {x : x ∉ A ∩ B} or, A △ B = {x : [x ∈ A and x ∉ B] or [x ∈ B and x ∉ A]} (Math Only Math, 2019)
The complement of a set A, denoted by Ac or A¯, is the set of all elements that are in the universal set S but are not in A. (Probability Course, 2020)
References
- • Contributor, T. (2005, April 5). Venn diagram. WhatIs.Com. https://whatis.techtarget.com/definition/Venn-diagram
- • Duignan, B. (2020, July 31). John Venn. Encyclopedia Britannica. https://www.britannica.com/biography/John-Venn
- • Kenton, W. (2020, January 17). How to Use a Venn Diagram. Investopedia. https://www.investopedia.com/terms/v/venn-diagram.asp
- • Math Only Math. (2019). Symmetric Difference using Venn Diagram. https://www.math-only-math.com/symmetric-difference-using-Venn-diagram.html
- • Probability Course. (2020). Set Operations. https://www.probabilitycourse.com/chapter1/1_2_2_set_operations.php
- • The Editors of Encyclopaedia Britannica. (2014, May 24). Set. Encyclopedia Britannica. https://www.britannica.com/topic/set-mathematics-and-logic