Venn Diagrams - Algebra of sets

Venn Diagrams - Algebra of sets

Introduction to Venn Diagrams

  • Venn Diagrams are a visual tool used in probability to illustrate the relationships between sets of items.
  • They are typically drawn as circles within a rectangle, where the rectangle represents the universal set - all possible outcomes - and each circle represents a particular set or group of outcomes.
  • The sets might overlap, indicating that some outcomes belong to both sets - these shared outcomes are called an intersection.
  • Outcomes that belong to one set but not another are found within the corresponding circle but outside of the other(s).

Understanding Set Terms

  • A set is a collection of distinct objects, which we often refer to as elements or members.
  • A set with no members is called an empty set or a null set.
  • The universal set is a set containing all objects under consideration. In a Venn Diagram, it’s represented by the rectangle.
  • The intersection of sets, often denoted as ‘A ∩ B’, is a set of all elements that are common to both sets.
  • The union of sets, often noted as ‘A ∪ B’, is a set of all elements that are in A, or in B, or in both. It’s visually represented as everything within both circles in a Venn Diagram.

Algebra of Sets

  • The algebra of sets refers to the mathematical operations that can be performed on sets, such as union, intersection, and complement.
  • The union operation combines sets. The union of two sets includes all elements that are in either set or in both.
  • A set’s complement, often denoted as ‘A’ or ‘A^c’, includes all elements that are not in the specified set, but are in the universal set.
  • The intersection operation finds common elements in sets. The intersection of two sets includes all elements that are in both sets.
  • Be aware that the difference between two sets “A - B” includes elements that are in A but not in B.

Using Venn Diagrams for Problem Solving

  • Venn Diagrams can show you in a visual way, the union, intersection, difference, and complement of sets.
  • They can be used to answer questions like: “how many elements are in either set A or B?” (A ∪ B), “how many elements are in both set A and B?” (A ∩ B), “how many elements are in A and not in B?” (A - B) and “how many elements are not in A but are in the universal set?” (A’).
  • Remember when using Venn diagrams to solve problems to carefully count the number of elements in each region of the diagram.
  • Always cross check your final answer with the total number of outcomes. It should account for all the elements in the universal set.

Important to Remember

  • Overlapping of circles in Venn Diagrams isn’t accidental. It visually represents the intersection of sets.
  • The union, intersection, difference, and complement can also be visually represented on a Venn Diagram.
  • Venn diagrams are especially useful for solving probability problems involving two or three sets. However, they can become cluttered and difficult to read with more sets.
  • The notation used for algebra of sets - such as ‘∩’ for intersection and ‘∪’ for union - is a universal mathematical language, so it’s important to remember these symbols and what they stand for.