p.10
Logical Equivalence and Tautologies
What is a tautology?
A compound proposition where its truth values are always true.
p.9
Implication and Biconditional Statements
What is the biconditional statement of propositions p and q?
p if and only if q, denoted as p ↔ q.
p.14
Quantifiers: Universal and Existential
What does the symbol ∃ represent in logic?
The existential quantifier, denoting 'for some', 'there exist', or 'there is at least one'.
p.12
Propositional Functions and Predicates
What does it mean to instantiate a variable?
To give a variable a specific value.
p.8
Implication and Biconditional Statements
What is the converse of a conditional statement?
Reversing the hypothesis and the conclusion.
p.11
Logical Equivalence and Tautologies
What are the possible classifications of a compound proposition?
Tautology, contradiction, or contingency.
p.7
Truth Values and Truth Tables
When is the compound proposition p → q false?
It is false when p is true and q is false; otherwise, it is true.
p.8
Implication and Biconditional Statements
Is the converse always true if the original statement is true?
No, the converse is not always true just because the original statement is true.
p.9
Implication and Biconditional Statements
When is the biconditional proposition p ↔ q true?
When both p and q have the same truth values.
p.5
Truth Values and Truth Tables
What is the truth value of the proposition '3 or -5 is negative'?
True (because -5 is negative).
p.7
Implication and Biconditional Statements
What are some alternative expressions for p → q?
If p, then q; q if p; p implies q; q when p; p only if q; q is necessary for p.
What is a Compound Proposition?
A proposition formed by combining one or more atomic propositions using logical connectives.
p.14
Negating Quantifier Expressions
When is the negation ¬∀x P(x) true?
When there is an x for which P(x) is false.
When is the compound proposition p ∨ q false?
It is false when both p and q are false.
p.7
Implication and Biconditional Statements
What is the definition of implication in propositional logic?
If p and q are proposition variables, the implication of p and q is 'if p, then q', denoted as p → q.
p.2
Truth Values and Truth Tables
What is the truth value of the proposition 'Horses are bigger than cats'?
True or False, depending on the context.
p.1
Propositions and Their Types
What distinguishes an Atomic Proposition from other propositions?
Atomic propositions cannot be broken down into smaller propositions.
What is Logic?
The systematic study of the principles of valid reasoning and inference.
p.11
Logical Equivalence and Tautologies
What is the relationship between p → q and ¬p ∨ q?
They are logically equivalent.
p.12
Propositional Functions and Predicates
What is the truth set of a predicate P(x)?
The set of all elements of D that make P(x) true when substituted for x.
When is the exclusive or p ⊕ q true?
When exactly one of p and q is true.
p.1
Truth Values and Truth Tables
What is the truth value of a true proposition?
True (T), corresponding to 1 in digital circuits.
p.3
Negation, Conjunction, and Disjunction
What is the truth table for negation?
If p is True (T), ¬p is False (F); if p is False (F), ¬p is True (T).
When is the compound proposition p ∨ q true?
It is true if at least one of p or q is true.
p.1
Propositions and Their Types
What is an Atomic Proposition?
A proposition whose truth or falsity does not depend on any other proposition.
p.7
Truth Values and Truth Tables
What is the truth table for the implication p → q?
The truth values are: T T T, T F F, F T T, F F T.
p.1
Truth Values and Truth Tables
What is the truth value of a false proposition?
False (F), corresponding to 0 in digital circuits.
p.3
Negation, Conjunction, and Disjunction
What is the negation of the proposition 'The integer 10 is even'?
The integer 10 is not even.
p.15
Negation, Conjunction, and Disjunction
What is the negation of the statement 'There is a man taller than three meters'?
There is no man taller than three meters.
p.12
Propositional Functions and Predicates
What is the universe of discourse or domain D of a predicate variable?
The set of all values that may be substituted in place of the variable.
p.14
Quantifiers: Universal and Existential
What is an existential statement?
A statement of the form '∃x ∈ D such that P(x)', meaning there exists an element x in D such that P(x) is true.
p.7
Implication and Biconditional Statements
What are the terms used for p and q in the implication p → q?
p is called the antecedent (or hypothesis) and q is called the consequent (or conclusion).
p.5
Truth Values and Truth Tables
What is the truth value of the proposition '√2 or π is an integer'?
False (neither √2 nor π is an integer).
p.13
Quantifiers: Universal and Existential
What does the symbol ∀ represent in logic?
The universal quantifier, denoting 'for all', 'for each', and 'for every'.
p.14
Negating Quantifier Expressions
What are De Morgan’s Laws for Quantifiers?
Rules for negating quantifier expressions, stating ¬∀x P(x) is equivalent to ∃x ¬P(x) and ¬∃x P(x) is equivalent to ∀x ¬P(x).
p.15
Negation, Conjunction, and Disjunction
What is the negation of the statement 'Every student in Discrete Mathematics class has taken Mathematics Logic'?
There exists at least one student in Discrete Mathematics class who has not taken Mathematics Logic.
p.6
Truth Values and Truth Tables
What is the exercise task related to truth tables?
Construct a truth table for the compound proposition (𝑝 ∧ 𝑞) ∨ ¬𝑟.
p.9
Implication and Biconditional Statements
When is the biconditional proposition p ↔ q false?
When p and q have opposite truth values.
p.1
Propositions and Their Types
What is a Proposition?
A declarative sentence that is either true or false, but not both.
p.2
Propositions and Their Types
Is 'Please do not fall asleep' a proposition?
No, it is a command, not a statement that can be true or false.
How do the truth values of p ⊕ q and p ↔ q relate?
They have opposite truth values.
What is the correct order of operations in logic?
¬ is performed first, then ∧ and ∨, and finally → and ↔.
p.2
Truth Values and Truth Tables
What is a truth table?
A table that gives the truth values of a compound proposition in terms of its component parts.
p.1
Propositional Functions and Predicates
What is a Propositional Function?
A statement that contains variables and becomes a proposition when the variables are replaced with specific values.
p.12
Propositional Functions and Predicates
What is the domain for Q(x): x is an animal?
{ cat, apple, computer, elephant }
p.12
Propositional Functions and Predicates
What is a predicate (or propositional function)?
A sentence that contains a finite number of predicate variables and becomes a statement when particular values are substituted for the variables.
p.5
Truth Values and Truth Tables
What is the truth value of the proposition '3 + 3 = 5 or 1 = 1'?
True (because 1 = 1 is true).
p.2
Propositions and Their Types
What is an atomic proposition?
A simple statement that can be true or false, such as 'The sky is blue.'
p.11
Logical Equivalence and Tautologies
What does p → q ↔ ¬p ∨ q signify?
It indicates that p → q is equivalent to ¬p ∨ q.
p.14
Quantifiers: Universal and Existential
When is an existential statement defined to be true?
If and only if P(x) is true for at least one x in D.
What is a compound proposition?
A proposition formed by combining one or more atomic propositions.
p.13
Quantifiers: Universal and Existential
What is a universal statement?
A statement of the form '∀x ∈ D, P(x)' means 'P(x) is true for all values of x in D'.
p.14
Negating Quantifier Expressions
When is the negation ¬∃x P(x) false?
When there is an x for which P(x) is true.
p.7
Implication and Biconditional Statements
What is the implication in the statement: 'If 1 + 1 = 3, then cats can fly'?
The implication is that the truth of '1 + 1 = 3' leads to the conclusion that 'cats can fly'.
p.13
Quantifiers: Universal and Existential
Translate 'For every x, if x is a natural number, then x is an integer' into logical symbolism.
∀x (if x ∈ ℕ, then x ∈ ℤ).
p.14
Quantifiers: Universal and Existential
When is an existential statement defined to be false?
If and only if P(x) is false for all x in D.
p.2
Propositions and Their Types
What is a simple statement in propositional logic?
A proposition represented by an atomic proposition variable.
p.8
Implication and Biconditional Statements
What is the inverse of a conditional statement?
Negating both the hypothesis and the conclusion.
p.7
Implication and Biconditional Statements
Determine the truth value of the statement: 'The moon is square only if the sun rises in the East.'
The truth value depends on the truth of the antecedent and consequent.
When is the compound proposition p ∧ q true?
It is true when both p and q are true; otherwise, it is false.
p.13
Quantifiers: Universal and Existential
What is a counterexample in the context of universal statements?
A value for x for which P(x) is false.
p.2
Propositions and Their Types
What type of statement is 'Math is fun'?
It is an opinion and not a proposition since it cannot be definitively true or false.
p.8
Logical Equivalence and Tautologies
What is the logical relationship between the contrapositive and the original statement?
The contrapositive is logically equivalent to the original statement.
p.14
Quantifiers: Universal and Existential
What does the expression ∀x ∃y P(x, y) mean?
For every x, there exists a y such that P(x, y) is true.
p.7
Implication and Biconditional Statements
Identify the hypothesis and conclusion in the statement: 'If you place your order by 11:59pm December 21st, then we guarantee delivery by Christmas.'
Hypothesis: You place your order by 11:59pm December 21st; Conclusion: We guarantee delivery by Christmas.
p.10
Logical Equivalence and Tautologies
What is a contingency?
A compound proposition that is neither a tautology nor a contradiction.
What is the conjunction of the propositions 'It is snowing' and 'I am cold'?
'It is snowing and I am cold.'
p.8
Logical Equivalence and Tautologies
Are the converse and inverse logically equivalent?
Yes, the converse and inverse are logically equivalent to each other.
What is the truth value of the proposition: 'The integer 2 is even but it is a prime number'?
False, because while 2 is even, it is not a prime number.
p.13
Quantifiers: Universal and Existential
When is a universal statement considered true?
It is true if and only if P(x) is true for every x in D.
p.1
Propositions and Their Types
Can all sentences be considered propositions?
No, not all sentences are propositions; only those that are declarative and can be true or false.
p.13
Quantifiers: Universal and Existential
Translate 'Every triangle is a polygon' into logical symbolism.
∀x (if x is a triangle, then x is a polygon).
p.13
Quantifiers: Universal and Existential
Translate 'All Sunway students are geniuses' into logical symbolism.
∀x (if x is a Sunway student, then x is a genius).
p.8
Implication and Biconditional Statements
Is the inverse guaranteed by the truth of the original statement?
No, the inverse is not guaranteed by the truth of the original statement.
p.13
Quantifiers: Universal and Existential
When is a universal statement considered false?
It is false if and only if P(x) is false for at least one x in D.
p.4
Truth Values and Truth Tables
What are the truth values for conjunction summarized in the truth table?
T T T, T F F, F T F, F F F.
p.10
Logical Equivalence and Tautologies
What is a contradiction?
A compound proposition where its truth values are always false.
p.8
Implication and Biconditional Statements
What is the contrapositive of a conditional statement?
Both reversing and negating the hypothesis and the conclusion.
p.10
Logical Equivalence and Tautologies
How can you determine if two compound propositions P and Q are logically equivalent?
By constructing the truth table P ↔ Q or using equivalence laws.
Determine the truth value of the proposition: '3 < 5 and 5 + 6 ≠ 11'.
False, because 5 + 6 = 11.
p.13
Propositional Functions and Predicates
What is the domain of R(x, y) in the given example?
Pairs (x, y) where x = 1, 2, or 3 and y = 3, 4.
p.2
Truth Values and Truth Tables
How do you determine the number of possible truth values based on the number of variables?
The number of possible truth values is 2^n, where n is the number of variables.
p.8
Implication and Biconditional Statements
Is the contrapositive always true if the original statement is true?
Yes, the contrapositive is always true if the original statement is true.
Determine the truth value of the proposition: '5 is positive and Kuala Lumpur is in Malaysia'.
True, both statements are correct.
p.8
Propositions and Their Types
What is the symbolic logic for the statement 'If it is snowing, then it is cold'?
p → q, where p is 'It is snowing' and q is 'It is cold'.
