Sets and Subsets in Mathematics

Sets and Subsets

• A set is a collection of objects.
• An element is an object in a set.
• The universal set contains all possible elements for a given sample.
• The number of elements in a set is denoted by n(set).
• A subset is a set whose elements are all part of another set.
• The complement of a set includes all elements from the universal set that are not in the set.
• The empty set is a set containing no elements.
• Disjoint sets have no elements in common.
• A finite set contains a countable number of elements.
• An infinite set contains an infinite number of elements.

Set Notation

• Sets are defined using curly braces.
• Subsets are represented using the symbol "⊂".
• The complement of a set is denoted by a prime symbol.

Venn Diagrams

• Venn diagrams show relationships between sets.
• Regions within the diagram represent different sets and subsets.
• The overlap between sets represents the intersection of those sets.

Exploring Relationships between Sets

• The intersection of two sets (A ∩ B) contains elements common to both sets.
• The union of two sets (A ∪ B) contains elements from either or both sets.
• The difference of two sets (A \ B) contains elements in set A but not in set B.

Example 1:

• Given a universal set S, subsets A and B are defined as multiples of 2 and 3, respectively.
• The Venn diagram shows the elements from S belonging to each set.
• The overlap represents elements in both A and B, or the intersection (A ∩ B).

Principle of Inclusion and Exclusion

• When sets have common elements, the number of elements in their union can be calculated using the formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B).
• When sets are disjoint, their intersection is empty, and the formula simplifies to: n(A ∪ B) = n(A) + n(B).

Example 2:

• Given sets A and B, the union (A ∪ B) includes all elements from both sets.
• The complement of the union (A ∪ B)' includes all elements from the universal set that are not in the union.
• Overcounting occurs when calculating the union's size (n(A ∪ B)) because common elements are counted twice.
• The principle of inclusion and exclusion compensates for this by subtracting the size of the intersection.

Example 3:

• In a class of 30 students, some eat breakfast and some eat a healthy lunch.
• Overlap in a Venn diagram represents students who eat both.
• To determine the number of students who eat a healthy lunch, we must consider those who eat both and those who eat only a healthy lunch.

Working with 3 Sets in a Venn Diagram

• When working with three sets, start with the intersection of all three sets.
• Numbers in all regions of the Venn diagram should add up to the total number in the universal set.
• Pay attention to wording: "only" means just that region.

Example 4:

• Given data about students studying different subjects, a Venn diagram can be used to visualize the relationships between the sets.
• By filling in the Venn diagram based on the given data, you can determine the number of students who meet specific criteria, such as studying only one subject or at least two subjects.

Example 5:

• A machine shop has employees with different skills.
• We can use a Venn diagram to visualize the employees who can operate different machines.
• By analyzing the diagram and using the principle of inclusion and exclusion, we can determine the total number of employees in the machine shop.