Set Theory Notes

Set Theory Study Notes

1.1 Introduction to Set Theory

  • Developed by George Boole (1815-1864) and Georg Cantor (1845-1918).
  • Sets have varied and useful applications in mathematics and everyday life.
  • The focus is on understanding sets in general.
1.1.1 Basic Symbols
  • E: stands for "belongs to" or "is in" or "member of"
  • |: stands for "such that"
  • ∃: stands for "there exists"
  • ∩: stands for "intersection"
  • ∪: stands for "union"
  • ⊆: stands for "is a subset of" or "is contained in"
  • ⊂: stands for "contains"
  • U: stands for "universal set"
  • Φ: stands for "null or empty set"
  • {}: represents the null or empty set
  • A': stands for "the complement of A"
  • a|b: stands for "a divides b"
  • R: stands for the set of real numbers
  • N: stands for the set of counting or natural numbers
  • Z: stands for the set of integers
  • Q: stands for the set of rational numbers
  • C: stands for the set of complex numbers
1.1.2 Set Notation
  • Lowercase letters (a, b, c, …) denote elements (members) of a set.
  • Capital letters (A, B, C, …) denote sets.
  • Elements are usually enclosed in braces.
1.1.3 Set Specification
  • Roster Method: Specify a set by listing its elements in braces.
    • Example: A = {2, 4, 6, 8}.
  • Set Builder Notation: Specify a set using defining properties.
    • Example: A = {x: x is even and 2 ≤ x ≤ 8}.
1.1.4 Definition
  • A set is a well-defined collection of distinct objects called elements.
  • Well-defined means it must be clear whether an object belongs to the set based on common properties.
Examples of Specifying Sets:

  1. Roster Method

    • A = {a, b, c, d, e, f,…, z} (lowercase letters)
    • B = {1, 2, 3, 4, 5,…, 10} (first ten counting numbers)
    • C = {2, 4, 6, 8, …, 12,…} (positive even counting numbers)
    • D = {2, x, c, v, b, n, m} (letters on the bottom row of a keyboard)
    • E = {Babalola, Aina, Ogunwemo} (names of Deans at Babcock University)

  2. Set Builder Notation

    • S = {x | x² = 9} yields S = {-3, 3}.
    • R = {x | x² - 4x + 3 = 0} yields R = {3, 1}.
    • H = {x | x is H.O.D. in Babcock University}
    • P = {x | x is a private university in Nigeria}
    • Q = {x | x is a positive number less than 100}
    • D = {x | x is a day of the week} = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}.

1.2 Properties of Sets

1.2.1 Set Inclusion
  • Let A and B be two sets.
  • A ⊆ B: A is a subset of B if every element of A is in B and at least one element in B is not in A (proper subset).
  • A ⊇ B: A is an improper subset if both sets are equal.
Examples:
  1. Let A = {a, b, c, d, e, f} and B = {a, b, c} ⇒ B ⊆ A.
  2. Let D = {z | z is a day of the week} and T = {z | z is a day whose first letter is t} ⇒ T ⊆ D.
  3. Given Z = {0, 1, 2, 3, ±4,…} and W = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,…} ⇒ W ⊆ Z.
  4. If X = {+1, +3} and Y = {x | x² - 4x + 3 = 0} ⇒ X ⊆ Y and Y ⊆ X.
  5. Let X = {x | x² - 3x + 2 = 0} and Y = {1, 2} ⇒ Y ⊆ X and X ⊆ Y.
1.2.2 Cardinality of a Set
  • The cardinality of a set is the number of elements in the set.
  • It is denoted by n(A) or o(A), where o(A) signifies the order of A.
Cardinality Examples:
  1. Let Y be the set of the Yoruba alphabet: n(Y) = 25.
  2. Let A = ∅: n(∅) = 0.
  3. Let E be the set of English alphabet: n(E) = 26.
  4. Let S be the set of states in Nigeria: n(S) = 36.
Cardinality of Infinite Sets
  • The cardinality of a countably infinite set is $ ext{Aleph-null}$ (ℵ₀).
  1. Let N be the set of natural numbers: n(N) = ℵ₀.
  2. Let ZZ be the set of integers: n(ZZ) = ℵ₀.
1.3 Special Sets

  1. Universal Set: Denoted by U, it contains all elements under consideration in a given problem.

    • The universal set may change from problem to problem.
    • Example: If the elements of a set are taken from N, then U = N = {1, 2, 3, 4, …}.

  2. Empty or Null Set: Denoted by {}, it contains no elements and is contained in every set.

    • Example of empty sets: the set of humans with 3 legs, set of dogs with horns, etc.

  3. Singleton Set: A set that contains only one element.

    • Examples: A = {0}, B = {x}.
1.4 Set Relations
1.4.1 Equivalent Sets
  • Two sets A and B are equivalent if they have the same cardinality, denoted A ~ B.
  • The elements of equivalent sets can be put in a one-to-one correspondence.
Infinite Sets
  • Infinite sets can also be equivalent. Example: The set of natural numbers N and the set of integers ZZ are equivalent.
1.4.2 Equal Sets
  • Two sets A and B are equal if they contain exactly the same elements: A = B.
    • Example: A = {1, 5, 6} and B = {5, 6, 1} ⇒ A = B.
1.4.3 Subset of a Set
  • Recall A ⊆ B if and only if every element of A is in B, and there exists at least one element of B not in A (proper subset).
  • If A ⊆ B, then it’s possible that A = B (improper subset).
1.4.3.1 Subsets of Finite Sets
  • Example: For U = {a, b}, the possible subsets are: {∅, {a}, {b}, {a, b}}.
  • For U = {a, b, c}, subsets include: {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}.
1.4.3.2 Subsets of an Infinite Set
  • The number of subsets of an infinite set cannot be computed because there is no end.
1.5 Set Difference
1.5.1 Definition
  • The difference between two sets A and B, denoted A - B, includes elements in A that are not in B:
    A−B=extx:xEA∧x<br/>∉BA - B = ext{ {x : x E A} \land {x <br />\notin B}}.
Examples:
  1. Let A = {a, b, c, d}, B = {a, b, c} ⇒ A - B = {d}.
  2. Let A = {a, b, c, d, e, f, 2, z}, B = {a, b, c, d, e, f, g, 2, 8} ⇒ B - A = {g, 8}.
1.5.2 Symmetric Difference
  • The symmetric difference between sets A and B is denoted A Δ B:
    C = (A - B)  (B - A).
Examples:

Let A = {1, 2, 3, 4}, B = {2, 4, 6, 8}.

  1. (i) A - B = {1, 3}
  2. C - A = {5, 6}, etc.
1.6 Complement of a Set
  • The complement of a set A, denoted A' or A^c, contains all elements that are in the universal set U but not in A:
    A′=x:x<br/>∉A.A' = { x : x <br />\notin A} .
Examples:
  1. Let U = {1, 2, …, 10} and A = {1, 4, 5, 6, 9} ⇒ A' = {2, 3, 7, 8, 10}.
  2. Let U = N and A = 2N. Then A' = {x: x is odd}.
1.7 Other Operations on Sets
1.7.1 Union of Sets
  • The union of sets A and B is denoted A ∪ B:
    A  B = { x : x ext{ is in } A ext{ or } B}.
Examples:
  1. Let A = {1, 2, 3, 5, 6} and B = {0, 1, 3, 7, 11} ⇒ A ∪ B = {0, 1, 2, 3, 5, 6, 7, 11}.
Properties of Union of Sets:
  • (a) Commutativity: A ∪ B = B ∪ A.
  • (b) Associativity: (A ∪ B) ∪ C = A ∪ (B ∪ C).
1.7.2 Intersection of Sets
  • The intersection of A and B is denoted A ∩ B:
    A  B = { x : x ext{ is in both } A ext{ and } B}.
Examples:
  1. A = {1, 2, 3, 4, 5} and B = {0, 1, 4, 8, 11} ⇒ A ∩ B = {1, 4}.
Properties of Intersection of Sets:
  • (a) Commutativity: A ∩ B = B ∩ A.
  • (b) Associativity: (A ∩ B) ∩ C = A ∩ (B ∩ C).
1.8 Basic Axioms in the Algebra of Sets
  1. Commutative Axioms
    • Commutativity with respect to Union: A ∪ B = B ∪ A.
    • Commutativity with respect to Intersection: A ∩ B = B ∩ A.
  2. Associative Axioms
    • Union: A ∪ (B ∪ C) = (A ∪ B) ∪ C.
    • Intersection: A ∩ (B ∩ C) = (A ∩ B) ∩ C.
  3. Distributive Axioms
    • A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
    • A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
  4. Idempotent Axioms
    • A ∪ A = A.
    • A ∩ A = A.
  5. Absorption Axioms
    • A ∪ (A ∩ B) = A.
    • A ∩ (A ∪ B) = A.
  6. Axioms of Complementation
    • A ∪ A' = U.
    • A ∩ A' = φ.
  7. Axiom of Double Complementation
    • (A')' = A.
  8. Axioms of De Morgan
    • (A ∪ B)' = A' ∩ B'.
    • (A ∩ B)' = A' ∪ B'.
1.8.1 Simplifying Expressions Involving Sets
  • You can simplify expressions using set axioms:
  1. Example: Show that A ∪ (A' ∩ B) = A ∪ B.
  2. Another Example: Simplify (A ∩ B) ∪ (A ∩ B') ∪ (A' ∩ B) ∪ (A' ∩ B').
1.9 Venn-Euler Diagrams
  1. Definition: A Venn-Euler diagram is a pictorial representation of relationships involving sets, often shown with overlapping circles.
  2. Applications: Used to illustrate set operations such as Union, Intersection, and Set Difference.
1.10 Exercises
  1. Express each set in set-builder form:
    • (1) {1,2,3,4,5,6}
    • (2) {2,4,6,8,10}
    • (3) {1,3,5,7,9}
    • (4) {a, c, e, i, o, u}
  2. Determine whether the following sets are finite or infinite:
    • (1) A = {x: x is a multiple of 5}
    • (2) B = {x: x is an African in the World}
    • (3) C = {y: y is a number between 1 and 30}
  3. Specify sets by listing their elements:
    • (1) integers that are multiples of 4 but less than 40.
    • (2) natural numbers that are multiples of three.
    • (3) positive numbers less than -1.