Logic Fundamentals: Negation, Conjunction, Disjunction

Questions and Answers

What does the conjunction of two propositions represent?

It combines propositions with 'or'.

It is false when both propositions are false.

It is true if at least one proposition is true.

It is true only when both propositions are true. (correct)

A disjunction is false when both propositions are true.

False

What is the negation of the proposition 'Vandana’s smartphone has at least 32GB of memory'?

Vandana’s smartphone has less than 32GB of memory.

The conditional statement p → q is false when p is true and __________ is false.

q Signup and view all the answers

Match the logical operations with their definitions:

Conjunction = True when both propositions are true Disjunction = True when at least one proposition is true Negation = Inverts the truth value of a proposition Conditional = True unless true implies false Signup and view all the answers

Under which conditions is the statement p → q true?

Both A and B Signup and view all the answers

A biconditional statement p ↔ q is false if p and q have the same truth values.

False Signup and view all the answers

What is the contrapositive of the statement p → q?

not q implies not p Signup and view all the answers

A statement that is neither a tautology nor a contradiction is called a __________.

contingency Signup and view all the answers

Match the logical operations with their definitions:

Negation = The opposite truth value of a proposition Conjunction = True if both propositions are true Disjunction = True if at least one proposition is true Conditional = True unless a true antecedent leads to a false consequent Signup and view all the answers

Study Notes

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.

Description

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.