Propositional Logic and Quantifiers

Propositional Logic

• Truth Tables: Truth tables are used to demonstrate the truth values of propositions.
• Logical Equivalence: The equivalence of two propositions means they have the same truth values in all possible cases.
• Contrapositive Property: The contrapositive of a statement has the same truth value as the original statement.
• Negation: The negation of a proposition is the opposite of the original proposition (e.g., if the original is true, the negation is false).
• Logical Connectives: Logical connectives such as AND (∧), OR (∨), and NOT (¬) are used to combine and modify propositions
• Conditional Statements: Conditional statements are in the form "If p then q." These statements are only false when p is true and q is false.

Quantifiers

• Universal Quantifier (∀): The universal quantifier means "for all" or "for every".
• Existential Quantifier (∃): The existential quantifier means "there exists" or "there is at least one".
• Negation of Quantified Statements: To negate a quantified statement, you change the quantifier and negate the proposition. For example, the negation of "∀x P(x)" is "∃x ¬P(x)".

Proof Techniques

• Proof by Induction: This method is used to prove statements that are true for all natural numbers. It involves proving the base case and then proving the inductive step.
• Proof by Contraposition: This method is used to prove a statement by proving the contrapositive of that statement, which has the same truth value.

### Mathematical Induction - Specific Examples

• Summation of Natural Numbers: The sum of the first n natural numbers can be expressed as n(n+1)/2.
• Summation of Cubes: The sum of the first n cubes can be expressed as [(n(n+1))/2]².
• Summation of Reciprocals of Fourth Powers: The sum of the reciprocals of the fourth powers of the first 2n natural numbers can be expressed as (1/3)[1 – 1/(n + 1)]⁴.

Number Theory

• Even Numbers: A number is considered even when it is divisible by 2.
• **Proof by Contraposition: ** The contrapositive of the statement "If n² is even, then n is even" is equivalent to "If n is odd, then n² is odd". This is often easier to prove.