Essential Set Theory Notations to Know for Discrete Mathematics

Set theory notations are essential in Discrete Mathematics, helping us understand collections of distinct objects. These notations clarify relationships between sets, their elements, and operations like union, intersection, and subsets, forming the foundation for more complex concepts.

1. Set notation: { }

   • Represents a collection of distinct objects or elements.
   • Elements are enclosed within curly braces.
   • Order of elements does not matter; duplicates are not allowed.

2. Element of: ∈

   • Indicates that an object is a member of a set.
   • Example: If A = {1, 2, 3}, then 2 ∈ A.
   • Used to express relationships between sets and their elements.

3. Not an element of: ∉

   • Indicates that an object is not a member of a set.
   • Example: If A = {1, 2, 3}, then 4 ∉ A.
   • Helps clarify which elements belong to a set and which do not.

4. Subset: ⊆

   • Indicates that all elements of one set are contained within another set.
   • Example: If A = {1, 2}, then A ⊆ {1, 2, 3}.
   • A set can be a subset of itself.

5. Proper subset: ⊂

   • Indicates that one set is a subset of another but not equal to it.
   • Example: If A = {1, 2}, then A ⊂ {1, 2, 3}.
   • A proper subset must have fewer elements than the set it is compared to.

6. Union: ∪

   • Combines all elements from two or more sets, removing duplicates.
   • Example: A ∪ B = {1, 2} ∪ {2, 3} = {1, 2, 3}.
   • Represents the total collection of elements from the involved sets.

7. Intersection: ∩

   • Represents the common elements shared between two or more sets.
   • Example: A ∩ B = {1, 2} ∩ {2, 3} = {2}.
   • Useful for identifying overlapping elements.

8. Set difference: \

   • Represents the elements in one set that are not in another.
   • Example: A \ B = {1, 2} \ {2, 3} = {1}.
   • Helps in distinguishing unique elements of a set.

9. Complement: A'

   • Represents all elements not in set A, relative to a universal set.
   • Example: If U = {1, 2, 3, 4} and A = {1, 2}, then A' = {3, 4}.
   • Useful for understanding the elements outside a specific set.

10. Empty set: ∅ or { }

    • A set that contains no elements.
    • Denotes the absence of any objects.
    • Important in set theory as a foundational concept.

11. Universal set: U

    • Represents the set that contains all possible elements relevant to a particular discussion.
    • All other sets are subsets of the universal set.
    • Context-dependent; varies based on the problem at hand.

12. Cartesian product: ×

    • Represents the set of all ordered pairs from two sets.
    • Example: A = {1, 2} and B = {x, y}, then A × B = {(1, x), (1, y), (2, x), (2, y)}.
    • Useful in creating relationships between two sets.

13. Power set: P(A)

    • The set of all possible subsets of a set A, including the empty set and A itself.
    • Example: If A = {1, 2}, then P(A) = {∅, {1}, {2}, {1, 2}}.
    • The number of subsets is 2^n, where n is the number of elements in A.

14. Cardinality: |A|

    • Represents the number of elements in a set A.
    • Example: If A = {1, 2, 3}, then |A| = 3.
    • Important for comparing the sizes of different sets.

15. Set builder notation: {x | P(x)}

    • A concise way to describe a set by stating the properties that its members must satisfy.
    • Example: {x | x > 0} represents the set of all positive numbers.
    • Useful for defining sets with specific criteria.