Introduction to Sets

Module 6: The Language of Sets

• Sets are fundamental building blocks in various mathematical disciplines like Graph Theory, Abstract Algebra, Real Analysis, Complex Analysis, Linear Algebra, and Number Theory.
• In mathematics, a set is a collection of distinct objects.
• Distinct objects means no duplication of elements within a set. Elements are listed only once.
• Well-defined objects means it is possible to clearly determine whether any given object is or is not in the set.

Examples of Well-Defined vs. Not Well-Defined Sets:

• Well-defined: The set of female presidents of the Philippines, the set of quadrilaterals, the set of rainbow colors, the set of even numbers less than 80, and the set of ASEAN member countries.
• Not Well-defined: The set of good Filipino writers, the set of best books in the library, the set of difficult subjects in SHS, the set of delicious smoothies, and the set of smart people in the meeting.

Rules for Writing Sets:

• Set names should be in capital letters (A, B, C, ..., X, Y, Z).
• Elements should be in lowercase letters (a, b, c, ..., x, y, z).
• Enclose elements within braces { }.
• If an object is an element of the set, use the symbol ∈, for example: f ∈ A.
• If an object is not an element of the set, use the symbol ∉, for example: a ∉ A.

The Set of Real Numbers:

• Natural Numbers (N): Counting numbers (positive integers) like 1, 2, 3, ...
• Integers (Z): Natural numbers, their negatives, and zero such as ...-4, -3, -2, -1, 0, 1, 2, 3, 4, ...
• Rational Numbers (Q): Numbers represented as a/b (where a and b are integers and b ≠ 0) with terminating or repeating decimal representations like -15, -2, 0, 23, -1/4, 3/7, 15/2, -2.75, 1.625, -0.3333, 5.272727.
• Irrational Numbers (Q'): Numbers represented as non-repeating and non-terminating decimals such as √2 = 1.414213562 ..., π = 3.141592654 ...
• Real Numbers (R): Includes both rational and irrational numbers.

Methods of Writing a Set:

• Enumeration or Roster Method: List elements in any order without repetition, enclosed by curly braces, for example: A = {f,r, e, s,h,m, n}.
• Defining or Rule Method: Define the set by stating the common properties of its members, using set-builder notation like {x | P(x)} or {x: P(x)}.

Kinds of Sets:

• Finite Set: Limited or countable elements, e.g., A = {1, 2, 3, ..., 16}, with cardinality n(A) = 16.
• Infinite Set: Unlimited or uncountable elements, e.g., W = {0, 1, 2, 3, ...}
• Unit Set (Singleton Set): Contains only one element, e.g., F = {11}, with cardinality n(F) = 1.
• Empty Set (Null Set): Contains no objects or elements, denoted by {} or Ø, e.g., H = {}, with cardinality n(H) = 0.
• Equal Sets: Sets containing the same elements, symbolically A = B.
• Equivalent Sets: Sets with the same number of elements or cardinality, denoted by A ≈ B.
• Disjoint Sets: Two sets having no common elements are disjoint.
• The infinity of a set is denoted by three dots (...), known as an ellipsis, indicating "and so forth".

Subsets, Supersets, and Power Sets:

• Subset: A is a subset of B (A ⊆ B) if every element of A is contained in B.
• The total subsets of a given set of "n" elements is 2^n.
• Proper Subset: Every element of A is in B, but B contains at least one element not in A, symbolized as A ⊂ B.
• A set consisting of n elements has 2^n - 1 proper subsets.
• Superset: If A is a subset of B, B is a superset of A, denoted by B ⊇ A.
• Power Set: The set of all subsets of A, denoted by P(A).
• The cardinality of the power set is |P(A)| = 2^n.
• The empty set is a subset of every set.

Universal Set and Complementary Sets:

• Universal Set: Contains all possible elements under consideration.
• Complementary Sets: Sets A and B are complementary if they share no common elements, and their union is the universal set.
• The complement of A is denoted by A^c.

Union and Intersection of Sets:

• Union (A ∪ B): Set containing all elements from both sets.
• Intersection (A ∩ B): Set containing the common elements of both sets.

Product Sets:

• The product set of two non-empty sets A and B is the set of all ordered pairs (a, b) where a is from set A, and b is from set B.
• The symbol "x" is used for the operation.
• A x B ≠ B x A

Module 7: Elementary Logic

• Logic deals with the methods of reasoning.
• Aristotle pioneered logical reasoning.
• Logic helps steer us in the direction of truth.

Propositions, Simple and Compound Statements:

• A proposition (or statement) is a declarative sentence that is either true or false, but not both.
• Propositions are represented by lowercase letters (p, q, r, s, ..., z)
• The table shows examples of propositions and sentences that are not propositions.

Examples of Propositions:

• Tarlac is a province in Region III.
• Elephants have wings.
• 15 – 2x = 8, if x = 5.
• Twenty is less than fifty.
• 5² + 1 is a prime number.

Examples of sentences that are not Propositions:

• Where is your teacher?
• Open the door.
• Ouch!
• This food is delicious.
• 5x + 2y = 7 (with no assigned values).

Proposition Types:

• Simple statement: A single statement without other statements as parts.
• Compound statement: Contains two or more statements joined by logical connectives.

Logic Connectives and Symbols:

• Negation: "not," symbol: ~ or ¬, symbolic form: ~p, read as: "not p".
• Conjunction: "and/but", symbol: ∧, symbolic form: p ∧ q, read as "p and q".
• Disjunction: "or", symbol: ∨, symbolic form: p ∨ q, read as "p or q".
• Conditional/Implication: "if..., then", symbol: →, symbolic form: p → q, read variations like "if p, then q","p implies q" etc.
• Biconditional: "if and only if", symbol: ↔, symbolic form: p ↔ q, read variations like "p if and only if q","p implies q and q implies p".

Compound Proposition Elements:

• In "if p, then q", p is the antecedent (hypothesis or premise).
• q is the consequent (or conclusion).

Module 8: Truth Tables

• The truth value is either true (T) or false (F) for a simple statement.
• The truth value for compound statements depends on its constituent simple statements and connectives.
• A truth table shows the truth value of a compound statement for all possible truth values of its simple statements.

Truth Tables for Propositional Logic:

• Negation: If a statement is true, its negation is false. If a statement is false, its negation is true.
• Conjunction: p ∧ q is TRUE only when both p and q are true.
• Disjunction: p ∨ q is TRUE if at least one of p or q is true.
• Conditional: p → q is FALSE when p is true and q is false; otherwise, it’s true.
• Biconditional: p ↔ q is TRUE when p and q have the same truth value.

Statement Types:

• Tautology: A proposition that is always true in all possible cases.
• Contradiction: A proposition that is always false in all possible cases.
• Contingency: A proposition whose truth values are not always true nor always false.

Module 9: Statements Related to Conditional Statements and Logical Equivalence

• Logical Equivalence: Two statements with the same truth values in all possible cases.
• Symbolically, p ⇔ q (read as p and q are logically equivalent).

Conditional Statement Relationships:

• Converse: Formed by interchanging the antecedent p and the consequent q. If p →q, the converse is q →p.
• Inverse: Formed by negating both the antecedent p and the consequent q. If p →q, the inverse is ~p→~q.
• Contrapositive: Formed by negating both p and q and then interchanging them. If p →q, the contrapositive is ~q→~p.

Key Equivalences:

• A conditional statement is equivalent to its contrapositive: p → q ≡ ~q →~p.
• The converse of a statement is equivalent to its inverse: q → p ≡ ~p →~q.