Key Set Theory Notation to Know for Mathematical Logic

Set theory notation is essential in mathematical logic, providing a clear way to describe collections of objects. Understanding symbols like ∈, ⊆, and ∪ helps express relationships between sets, making complex ideas easier to grasp and analyze.

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.
   • For 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.
   • For 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 also elements of another set.
   • For example, if A = {1, 2} and B = {1, 2, 3}, then A ⊆ B.
   • A set is always a subset of itself.

5. Proper subset: ⊂

   • Indicates that one set is a subset of another but not equal to it.
   • For example, if A = {1, 2} and B = {1, 2, 3}, then A ⊂ B.
   • 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.
   • For example, if A = {1, 2} and B = {2, 3}, then A ∪ B = {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.
   • For example, if A = {1, 2} and B = {2, 3}, then A ∩ B = {2}.
   • Useful for identifying overlapping elements in sets.

8. Set difference: \

   • Represents the elements in one set that are not in another set.
   • For example, if A = {1, 2, 3} and B = {2, 3}, then A \ B = {1}.
   • Helps in distinguishing unique elements of a set.

9. Complement: A^c or A'

   • Represents all elements not in a given set, relative to a universal set.
   • For example, if U = {1, 2, 3, 4} and A = {1, 2}, then A^c = {3, 4}.
   • Important for understanding the relationship between a set and the universal set.

10. Empty set: ∅ or { }

    • Represents a set with no elements.
    • Denotes the absence of any members.
    • Fundamental concept in set theory, often used in proofs.

11. Universal set: U

    • Represents the set that contains all possible elements relevant to a particular discussion.
    • Every other set is a subset of the universal set.
    • Context-dependent; its elements vary based on the problem at hand.

12. Cartesian product: ×

    • Represents the set of all ordered pairs from two sets.
    • For example, if A = {1, 2} and B = {x, y}, then A × B = {(1, x), (1, y), (2, x), (2, y)}.
    • Useful in defining relations and functions.

13. Power set: P(A)

    • Represents the set of all possible subsets of a set A, including the empty set and A itself.
    • If A has n elements, then P(A) has 2^n elements.
    • Important for combinatorial problems and understanding set relationships.

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

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

15. Cardinality: |A|

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