Logic Fundamentals: Negation, Conjunction, Disjunction

Negation

• Negation transforms statements into their opposite meaning.
• Example: "Vandana’s smartphone has at least 32GB of memory" negates to "Vandana’s smartphone has less than 32GB of memory."

Conjunction ("And")

• Conjunction of propositions p and q is denoted as p ∧ q.
• It is true only when both p and q are true.
• Example: p: "Rebecca’s PC has more than 16 GB free hard disk space" and q: "The processor runs faster than 1 GHz" leads to p ∧ q: "Rebecca’s PC has more than 16 GB free hard disk space and the processor runs faster than 1 GHz."

Disjunction ("Or")

• Disjunction of propositions p and q is denoted as p ∨ q.
• It is false only when both p and q are false.
• Example: For p and q same as above, p ∨ q states: "Rebecca’s PC has at least 16 GB free hard disk space, or the processor runs faster than 1 GHz."

Exclusive Or

• Exclusive or (XOR) denoted by p ⊕ q is true when exactly one of p or q is true.

Conditional Statements

• A conditional statement p → q expresses "if p, then q."
• It is false only when p is true and q is false.
• If p is false, then the statement is true regardless of q's value.
• Different expressions of conditional statements include: "p implies q," "q if p," and "p is sufficient for q."

Biconditionals

• Biconditional statement p ↔ q is true when both have the same truth values.
• Also known as bi-implications, expressed as "p if and only if q."
• Equivalent expression: p ↔ q ≡ (p → q) ∧ (q → p).

Propositional Equivalences

• Tautology: A compound proposition always true regardless of variables.
• Contradiction: A compound proposition always false.
• Contingency: A compound proposition that is neither tautology nor contradiction.

Logical Equivalences

• Compound propositions with identical truth values in all scenarios are logically equivalent.
• Demonstrated equivalence: ¬(p ∨ q) is logically equivalent to ¬p ∧ ¬q.

Negating Quantified Expressions

• Negation of universal quantification (∀x P(x)) is expressed as there exists an x such that ¬P(x), represented as ∃x ¬P(x).
• Example: "Every student has taken calculus" negates to "There is a student who has not taken calculus."

Examples of Negating Quantifiers

• Negation of ∀x(x² > x) converts to ∃x(x² ≤ x).
• Negation of ∃x(x² = 2) converts to ∀x(x² ≠ 2).

Translating Statements into Logical Expressions

• "Every student in this class has studied calculus" can be expressed with predicates and quantifiers.

Explore the essential concepts of logic through a quiz focusing on negation, conjunction, disjunction, and conditional statements. Test your understanding of how these logical operations interact with propositions and their meanings. Perfect for students looking to deepen their grasp of logical reasoning.