Midterm_notes

What is the disjunction of statements P and Q?

The disjunction, denoted by P v Q, is true if at least one of P or Q is true.

What is the conjunction of the statements P and Q?

The conjunction, denoted by P ^ Q, is true if both P and Q are true.

What does the notation P ⇒ Q represent?

P ⇒ Q represents the implication 'If P, then Q'.

When is an implication P ⇒ Q false?

An implication P ⇒ Q is false when P is true and Q is false.

What is the biconditional statement P = Q?

The biconditional states that P is true if and only if Q is true.

What are logical connectives?

Symbols like ~, V, ^, ⇒, that combine one or more component statements.

What is a tautology?

A compound statement that is true for all possible combinations of truth values.

What is a contradiction?

A compound statement that is false for all possible combinations of truth values.

What does the universal quantifier denote?

The universal quantifier denotes 'for every' or 'for all' in quantified statements.

What is the existential quantifier?

The existential quantifier denotes 'there exists' or 'there is' in quantified statements.

What is a trivial proof?

A proof that shows a statement is true because the conclusion is true regardless of the premise.

What is a vacuous proof?

A proof that shows a statement is true because the premise is false for all cases.

What are De Morgan's Laws?

Laws that state: 1) ~(P V Q) = ~P ^ ~Q; 2) ~(P ^ Q) = ~P V ~Q.

What does the term 'logical equivalence' mean?

Two statements are logically equivalent if they have the same truth values.

How do you determine if two statements are logically equivalent?

If they yield the same truth values for all possible combinations of their components.

What is an open sentence in logic?

An open sentence is a statement that contains one or more variables and becomes a proposition when the variables are replaced with specific values.

How do open sentences relate to implications?

Open sentences can form implications when combined with specific values for the variables, creating statements of the form 'If P(x), then Q(x),' which are assessed for their truth based on the assigned values.

What is the biconditional statement P = Q?

The biconditional states that P is true if and only if Q is true.

What is a direct proof?

A direct proof is a method of demonstrating the truth of a statement by a straightforward chain of logical deductions from established facts or definitions.

What is quantification in logic?

Quantification in logic refers to the use of quantifiers like universal (∀) and existential (∃) to express the extent to which a predicate applies to a subject or subjects.

What is an axiom in logic and mathematics?

An axiom is a statement or proposition that is accepted as true without proof, serving as a starting point for further reasoning and arguments.

What is a proposition in logic?

A proposition is a declarative statement that is either true or false, but not both.

When should a theorem be used instead of a proposition?

A theorem is used when the statement has been proven based on previously established statements and is generally regarded as universally true, while a proposition may represent a statement proposed for consideration that has not necessarily been proven.

What is a corollary in logic and mathematics?

A corollary is a statement that follows readily from a previously proven statement, often considered a result or consequence of a theorem.

What is a trivial proof?

A proof that shows a statement is true because the conclusion is true regardless of the premise.

What is a vacuous proof?

A proof that shows a statement is true because the premise is false for all cases.

What is an incorrect proof?

An incorrect proof is a demonstration that fails to establish the truth of a statement due to logical errors or invalid reasoning.

What does the symbol ℕ represent?

The symbol ℕ represents the set of natural numbers, which includes all positive integers starting from 1: {1, 2, 3, ...}.

What does the symbol ℤ represent?

The symbol ℤ represents the set of integers, which includes all whole numbers, both positive and negative, as well as zero: {..., -3, -2, -1, 0, 1, 2, 3, ...}.

What does the symbol ℚ represent?

The symbol ℚ represents the set of rational numbers, which includes all numbers that can be expressed as the quotient of two integers (a/b), where b ≠ 0.

What does the symbol ℝ represent?

The symbol ℝ represents the set of real numbers, which includes all rational and irrational numbers.

What does the symbol ℂ represent?

The symbol ℂ represents the set of complex numbers, which includes all numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit.

What does the notation |x| represent?

The notation |x| represents the absolute value of x, which is the non-negative value of x without regard for its sign.

What does the notation f(x) imply in mathematics?

The notation f(x) denotes a function named f that takes an input x and produces an output, often representing a mathematical relationship.

What does the summation symbol Σ signify?

The summation symbol Σ represents the sum of a series of numbers, typically used to add up terms of a sequence.

What does the notation ∀ denote in logical statements?

The notation ∀ denotes 'for all,' indicating that a statement applies to every member of a specified set.

What does the notation ∃ denote in logical statements?

The notation ∃ denotes 'there exists,' signifying that there is at least one member in a specified set for which the statement holds true.

What do parentheses ( ) indicate in mathematical expressions?

Parentheses ( ) indicate grouping in mathematical expressions, indicating the order in which operations should be performed.

What does the notation x ∈ S mean?

The notation x ∈ S means that element x is a member of the set S.

What does the notation x ∉ S mean?

The notation x ∉ S means that element x is not a member of the set S.

What are common reasons for a proof to be incorrect?

Common reasons include assumptions without justification, misapplication of rules, invalid deductions, or overlooking counterexamples.

How can one verify the correctness of a proof?

One can verify the correctness of a proof by checking each step for logical consistency, ensuring all assumptions are justified, and confirming that the conclusion follows from the premises.

Example Proof: Let P be 'x = 2' and Q be 'x^2 = 4'. Is the proof 'If P is true, then Q is true (P ⇒ Q)' a valid statement?

This proof is valid because if x indeed equals 2, squaring it will result in 4, thus making the implication tru