Sets and Subsets Quiz

Sets and Subsets

• A set is a well-defined collection of objects. Membership can be definitively determined.
• Sets are denoted by uppercase italic letters (e.g., A, B, C).
• Elements of a set are denoted by lowercase letters (e.g., a, b, c).
• The empty set is denoted by ∅ or {}.
• ∈ means "is an element of," ∉ means "is not an element of."
• Sets can be defined by specifying properties their elements must satisfy: A = {a | a is _______} means "the set A is comprised of elements a where a satisfies _______."

Set Notation and Examples

• Sets can be listed using enumeration: {1, 2, 3}.
• Examples of common sets:
  • Z⁺: Positive integers
  • N: Non-negative integers (positive integers and zero)
  • Z: Integers
  • Q: Rational numbers (numbers expressible as a/b, where a and b are integers and b ≠ 0)
  • R: Real numbers

Subsets

• A ⊆ B means "A is a subset of B." All elements of A are also in B.
• A ⊄ B means A is not a subset of B. This doesn't necessarily mean there are no shared elements.
• Every set is a subset of itself (A ⊆ A).
• The empty set is a subset of every set (∅ ⊆ A).
• If A ⊆ B and B ⊆ A, then A = B.

Venn Diagrams

• Developed by John Venn.
• A visual representation of set relationships; it shows the relationship between sets, not individual elements.

Theorems on Sets

• If A ⊆ B and B ⊆ C, then A ⊆ C (transitive property).
• If A ⊆ B and C ⊆ B, the relationship between A and C is indeterminate (could be A ⊆ C, C ⊆ A, or neither).

Set Cardinality and Power Sets

• A finite set A has n distinct elements; n is the cardinality of A, denoted |A|.
• An infinite set is a set that is not finite.
• The power set P(A) is the set of all subsets of A, including the empty set.

Subset Examples

• Vowels are a subset of the alphabet.
• Letters in "see" are a subset of letters in "yes," which is a subset of letters in "easy."
• Positive integers are a subset of integers, which are a subset of real numbers.