What is the inverse of the statement 'If p, then q'?

If not p, then not q.

What is a tautology?

A compound proposition where its truth values are always true.

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p.8

Implication and Biconditional Statements

If not p, then not q.

p.10

Logical Equivalence and Tautologies

A compound proposition where its truth values are always true.

p.9

Implication and Biconditional Statements

p if and only if q, denoted as p ↔ q.

p.11

Logical Equivalence and Tautologies

p and q.

p.14

Quantifiers: Universal and Existential

The existential quantifier, denoting 'for some', 'there exist', or 'there is at least one'.

p.12

Propositional Functions and Predicates

To give a variable a specific value.

p.8

Implication and Biconditional Statements

Reversing the hypothesis and the conclusion.

p.11

Logical Equivalence and Tautologies

Tautology, contradiction, or contingency.

p.7

Truth Values and Truth Tables

It is false when p is true and q is false; otherwise, it is true.

p.8

Implication and Biconditional Statements

No, the converse is not always true just because the original statement is true.

p.9

Implication and Biconditional Statements

When both p and q have the same truth values.

p.11

Logical Equivalence and Tautologies

p → q.

p.5

Truth Values and Truth Tables

True (because -5 is negative).

p.3

Logical Connectives

If or only if.

p.12

Propositional Functions and Predicates

False.

p.3

Negation, Conjunction, and Disjunction

¬p.

p.7

Implication and Biconditional Statements

If p, then q; q if p; p implies q; q when p; p only if q; q is necessary for p.

p.1

Logical Connectives

A proposition formed by combining one or more atomic propositions using logical connectives.

p.14

Negating Quantifier Expressions

When there is an x for which P(x) is false.

p.5

Logical Connectives

It is false when both p and q are false.

p.5

Truth Values and Truth Tables

True.

p.7

Implication and Biconditional Statements

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

True or False, depending on the context.

p.1

Propositions and Their Types

Atomic propositions cannot be broken down into smaller propositions.

p.3

Logical Connectives

Conjunction (And).

p.1

Introduction to Logic

The systematic study of the principles of valid reasoning and inference.

p.11

Logical Equivalence and Tautologies

They are logically equivalent.

p.3

Logical Connectives

→ (If…., then…).

p.12

Propositional Functions and Predicates

The set of all elements of D that make P(x) true when substituted for x.

p.9

Logical Connectives

When exactly one of p and q is true.

p.1

Truth Values and Truth Tables

True (T), corresponding to 1 in digital circuits.

p.3

Negation, Conjunction, and Disjunction

If p is True (T), ¬p is False (F); if p is False (F), ¬p is True (T).

p.5

Logical Connectives

It is true if at least one of p or q is true.

p.1

Propositions and Their Types

A proposition whose truth or falsity does not depend on any other proposition.

p.12

Propositional Functions and Predicates

{ x ∈ D | P(x) }

p.3

Negation, Conjunction, and Disjunction

False.

p.7

Truth Values and Truth Tables

The truth values are: T T T, T F F, F T T, F F T.

p.1

Truth Values and Truth Tables

False (F), corresponding to 0 in digital circuits.

p.3

Negation, Conjunction, and Disjunction

The integer 10 is not even.

p.15

Negation, Conjunction, and Disjunction

There is no man taller than three meters.

p.12

Propositional Functions and Predicates

The set of all values that may be substituted in place of the variable.

p.9

Truth Values and Truth Tables

False.

p.14

Quantifiers: Universal and Existential

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

p is called the antecedent (or hypothesis) and q is called the consequent (or conclusion).

p.5

Truth Values and Truth Tables

False (neither √2 nor π is an integer).

p.13

Quantifiers: Universal and Existential

The universal quantifier, denoting 'for all', 'for each', and 'for every'.

p.14

Negating Quantifier Expressions

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

There exists at least one student in Discrete Mathematics class who has not taken Mathematics Logic.

p.6

Truth Values and Truth Tables

¬𝑝 ∨ ~𝑞

p.6

Truth Values and Truth Tables

Construct a truth table for the compound proposition (𝑝 ∧ 𝑞) ∨ ¬𝑟.

p.3

Logical Connectives

¬

p.9

Implication and Biconditional Statements

When p and q have opposite truth values.

p.11

Logical Equivalence and Tautologies

¬p ∨ q.

p.3

Logical Connectives

Disjunction (Or).

p.1

Propositions and Their Types

A declarative sentence that is either true or false, but not both.

p.9

Logical Connectives

p ⊕ q.

p.2

Propositions and Their Types

No, it is a command, not a statement that can be true or false.

p.9

Logical Connectives

They have opposite truth values.

p.12

Propositional Functions and Predicates

False.

p.10

Logical Connectives

¬ is performed first, then ∧ and ∨, and finally → and ↔.

p.2

Truth Values and Truth Tables

A table that gives the truth values of a compound proposition in terms of its component parts.

p.1

Propositional Functions and Predicates

A statement that contains variables and becomes a proposition when the variables are replaced with specific values.

p.12

Propositional Functions and Predicates

{ cat, apple, computer, elephant }

p.5

Logical Connectives

p ∨ q.

p.12

Propositional Functions and Predicates

A sentence that contains a finite number of predicate variables and becomes a statement when particular values are substituted for the variables.

p.9

Truth Values and Truth Tables

True.

p.5

Truth Values and Truth Tables

True (because 1 = 1 is true).

p.2

Propositions and Their Types

A simple statement that can be true or false, such as 'The sky is blue.'

p.11

Logical Equivalence and Tautologies

It indicates that p → q is equivalent to ¬p ∨ q.

p.14

Quantifiers: Universal and Existential

If and only if P(x) is true for at least one x in D.

p.8

Implication and Biconditional Statements

If q, then p.

p.5

Truth Values and Truth Tables

True.

p.10

Logical Connectives

Parentheses ( ).

p.2

Logical Connectives

A proposition formed by combining one or more atomic propositions.

p.13

Quantifiers: Universal and Existential

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 there is an x for which P(x) is true.

p.7

Implication and Biconditional Statements

The implication is that the truth of '1 + 1 = 3' leads to the conclusion that 'cats can fly'.

p.13

Quantifiers: Universal and Existential

∀x (if x ∈ ℕ, then x ∈ ℤ).

p.10

Implication and Biconditional Statements

¬r → (¬p ∧ q).

p.14

Quantifiers: Universal and Existential

If and only if P(x) is false for all x in D.

p.2

Propositions and Their Types

A proposition represented by an atomic proposition variable.

p.8

Implication and Biconditional Statements

Negating both the hypothesis and the conclusion.

p.12

Propositional Functions and Predicates

True.

p.7

Implication and Biconditional Statements

The truth value depends on the truth of the antecedent and consequent.

p.4

Logical Connectives

It is true when both p and q are true; otherwise, it is false.

p.13

Quantifiers: Universal and Existential

A value for x for which P(x) is false.

p.2

Propositions and Their Types

It is an opinion and not a proposition since it cannot be definitively true or false.

p.8

Logical Equivalence and Tautologies

The contrapositive is logically equivalent to the original statement.

p.14

Quantifiers: Universal and Existential

For every x, there exists a y such that P(x, y) is true.

p.7

Implication and Biconditional Statements

Hypothesis: You place your order by 11:59pm December 21st; Conclusion: We guarantee delivery by Christmas.

p.4

Logical Connectives

It is denoted as p ∧ q.

p.10

Logical Equivalence and Tautologies

A compound proposition that is neither a tautology nor a contradiction.

p.4

Logical Connectives

'It is snowing and I am cold.'

p.10

Implication and Biconditional Statements

(q ∨ ¬r) → ¬p.

p.8

Logical Equivalence and Tautologies

Yes, the converse and inverse are logically equivalent to each other.

p.12

Propositional Functions and Predicates

Q(x): x is an animal.

p.2

Truth Values and Truth Tables

Yes, but it is false.

p.4

Logical Connectives

False, because while 2 is even, it is not a prime number.

p.13

Quantifiers: Universal and Existential

It is true if and only if P(x) is true for every x in D.

p.1

Propositions and Their Types

No, not all sentences are propositions; only those that are declarative and can be true or false.

p.8

Implication and Biconditional Statements

If not q, then not p.

p.13

Quantifiers: Universal and Existential

∀x (if x is a triangle, then x is a polygon).

p.13

Quantifiers: Universal and Existential

∀x (if x is a Sunway student, then x is a genius).

p.8

Implication and Biconditional Statements

No, the inverse is not guaranteed by the truth of the original statement.

p.13

Quantifiers: Universal and Existential

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

T T T, T F F, F T F, F F F.

p.10

Implication and Biconditional Statements

p → (¬q ∧ r).

p.10

Implication and Biconditional Statements

q ↔ ¬p.

p.10

Logical Equivalence and Tautologies

A compound proposition where its truth values are always false.

p.8

Implication and Biconditional Statements

Both reversing and negating the hypothesis and the conclusion.

p.10

Logical Equivalence and Tautologies

By constructing the truth table P ↔ Q or using equivalence laws.

p.4

Logical Connectives

False, because 5 + 6 = 11.

p.13

Propositional Functions and Predicates

Pairs (x, y) where x = 1, 2, or 3 and y = 3, 4.

p.2

Truth Values and Truth Tables

The number of possible truth values is 2^n, where n is the number of variables.

p.8

Implication and Biconditional Statements

Yes, the contrapositive is always true if the original statement is true.

p.4

Logical Connectives

True, both statements are correct.

p.8

Propositions and Their Types

p → q, where p is 'It is snowing' and q is 'It is cold'.

p.3

Logical Connectives

Connectives.

p.6

Truth Values and Truth Tables

(¬𝑝 ∨ 𝑞) ∧ (¬𝑟)

Study Smarter, Not Harder

Study Smarter, Not Harder