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.

Test your knowledge on propositional logic and quantifiers with this quiz. It covers key concepts like truth tables, logical equivalence, negation, and the use of logical connectives. Additionally, the quiz addresses universal and existential quantifiers to deepen your understanding of logic.