∃ x ∀ y (xy ≠ 1).
It indicates the negation of a statement involving nested quantifiers.
It is denoted by p ∧ q.
They allow for more complex statements about relationships between variables.
A sequence of statements that ends with a conclusion.
A declarative sentence that is true or false, but not both.
Either p is false or q is true.
If p = 'I will have salad for lunch.' and q = 'I will have steak for dinner.', then p ∧ q = 'I will have salad for lunch and I will have steak for dinner.'
It suggests a strong local presence with international recognition.
Constructing new logical equivalences.
It represents the logical conjunction (AND) of two propositions.
A propositional statement that is always true.
True.
A pattern that allows us to deduce a conclusion from a set of true premise statements.
¬ ∀ x P(x) ≡ ∃ x ¬P(x), meaning not every student in the class has taken a course in calculus.
An implication that is a tautology.
Sufficient.
If 'P implies Q' and 'Q' is false, then 'P' must also be false.
If they contain the same truth values in all corresponding rows of their truth tables.
p → q and ¬p ∨ q.
Every student in the class has taken a course in calculus.
It combines two propositions to form their logical exclusive or.
Statements that have the same truth value in every possible scenario.
¬p → ¬q.
The cat is on the roof.
True when p is false or both p and q are true; false only when p is true and q is false.
As ∀ x(x > 0 → P(x)).
Using dictionaries or translation software.
True when p is false or q is true; false only when both p is true and q is false.
p and q have the same truth value.
'The cat is not sleepy' or 'It is not the case that the cat is sleepy.'
Applications of Propositional Logic.
Boolean operators.
Logically equivalent.
∀ x (C(x) ∨ M(x))
Syukron Abu Ishaq Alfarozi.
If P is a sufficient condition for Q, then P being true guarantees Q is true.
If p is true, then q is also true.
Rules that dictate the valid steps in logical reasoning.
True.
A way to create propositions using propositional functions.
A branch of logic that deals with predicates and quantifiers.
False.
A rule of inference stating that if p is true and p implies q, then q is true.
Statements (propositions) and compound statements built from simpler statements using Boolean connectives.
p → q.
A predicate and a quantifier.
Negation.
A compound proposition that is false no matter what.
There exists a real number z such that for all real numbers x and for all real numbers y, it is true that x + y = z.
There exists a person y such that every person x relies upon y.
¬ ∀ x ∃ y (P(x,y) ∧ ∃ z R(x,y,z)) ≡ ∃ x ¬ ∃ y (P(x,y) ∧ ∃ z R(x,y,z)).
p ↔ q means ¬(p ⊕ q).
A compound proposition that is neither a tautology nor a contradiction.
The order does not affect the meaning.
Predicates and Quantifiers.
'x is sleeping' (where x is any subject).
By combining two or more terms to find results that include all of them.
The set {T, F}, where T is true and F is false.
Valid and invalid mathematical arguments.
If it is not sunny, then we will not go swimming.
Expressing Boolean operators in different forms.
A quantifier that is placed within the scope of another quantifier.
It suggests a strong local foundation with international recognition.
The domain for x consists of the students in this class.
Subject, predicate, object, etc.
p.
An educational institution or university.
There is a student in the class who has taken a course in calculus.
Maintaining the original meaning and context.
The extent to which a predicate is true over a range of elements.
Symbols that express the quantity of specimens in a domain that satisfy a given property.
Rules that dictate the valid steps in logical reasoning to derive conclusions from premises.
((premise 1) ˄ (premise 2)) → conclusion.
If p = 'I will earn an A in this course' and q = 'I will drop this course', then p ⊕ q means 'I will either earn an A or drop the course, but not both.'
q → p.
'is sleeping'.
∀xP(x).
Implication or 'if...then' relationship.
Because it excludes the possibility that both p and q are true.
If 'P or Q' is true and 'P' is false, then 'Q' must be true.
q could be either true or false.
The second De Morgan law.
If you study hard (p), then you will get a good grade (q).
Biconditional.
'not p and not q'.
If p = 'My car has a bad engine.' and q = 'My car has a bad carburetor.', then p ∨ q = 'My car has a bad engine, or my car has a bad carburetor.'
Searching for 'cats NOT dogs' to find results about cats without any mention of dogs.
A table that assigns a Boolean value to the set of all Boolean n-tuples.
For all x, there exists a y such that P(x, y) holds.
M(x) represents 'x has visited Mexico'.
True.
A propositional statement that can be either true or false depending on the truth values of its components.
A sentence that declares a fact, having some definite meaning, not vague or ambiguous.
It means 'For all x that are greater than zero, P(x).'
¬p or ҧp.
Sufficient.
Apply De Morgan's Law.
p ⊕ q.
T for true propositions and F for false propositions.
If 'P or Q' is true and 'P' is false, then 'Q' must be true.
Maintaining the original meaning and context.
Conjunction.
As ∃ x(x > 0 → P(x)).
It implies that either p is true or q is true, but not both.
A statement that can be either true or false.
There exists a person x such that x relies upon every person y.
It reverses the truth value of a proposition.
p = 'It is below freezing.' q = 'It is snowing.'
The precedence of quantifiers is higher than all operator logic.
Marla has taken a course in computer science.
No, it is not a proposition because it is an imperative statement and does not have a truth value.
(¬ p ∧ p) ∨ (¬ p ∧ ¬ q)
(¬ p ∧ ¬ q) ∨ F by the commutative law for disjunction.
Discrete Mathematics.
It will have 2^n rows.
A branch of logic that deals with propositions and their relationships.
ugm.ac.id.
It signifies that there exists an element x in the domain such that P(x) is true.
Universitas Gadjah Mada, Yogyakarta, Indonesia.
True.
By letters called propositional variables (p, q, r, s, etc.).
Ph.D. in Information Technology.
A statement that expresses an implication, typically in the form 'If P, then Q'.
From atomic propositions by applying propositional operators, e.g., ¬p, p ∧ q, (p ∧ ¬q) → q.
King Mongkut’s Institute of Technology Ladkrabang, Bangkok, Thailand.
For every person x, there exists a person y such that x relies upon y.
When P(x) is false for every x in the domain.
When one quantifier is within the scope of another quantifier.
The conclusion (or consequence).
Propositional Equivalences.
To derive conclusions from premises in a logical manner.
Yes, it is a proposition but it is false.
It narrows the search by including results that contain all specified terms.
(p ∨ q) → ¬r
It broadens the search to include results that contain any of the specified terms.
¬ p ∧ [¬ (¬ p) ∨ ¬ q]
Whether reasoning is correct.
Either of the elements in the Boolean domain (T or F).
It forms the foundations for computer circuits design and is used to build computer programs.
Aristotle.
p ≡ q.
If it is raining, then the ground is wet.
Raining tomorrow is a sufficient condition for me not going to town.
He is a faculty member in the Department of Electrical and Information Engineering at Universitas Gadjah Mada.
False.
By using a truth table to show that both expressions have the same truth values.
A branch of logic that deals with propositions and their relationships.
An atomic proposition is either a Boolean constant or a propositional variable, e.g., T, F, p.
The proposition 'if p, then q.'
¬ ∃ x P(x) ≡ ∀ x ¬P(x), meaning there is no student in the class who has taken a course in calculus.
Translation software or online translation services.
The object or entity that the sentence is about.
It represents the statement 'P(x) for all values of x in the domain.'
True.
Predicate calculus.
Aristotle.
If I don’t go to town then it will rain tomorrow.
Then q is true.
Exclusive Disjunction.
It becomes ∀ y ¬(P(x,y) ∧ ∃ z R(x,y,z)).
If P(x) is true for every x in the domain.
No, it is not a proposition because it is a question and does not have a truth value.
An operator that takes one operand (e.g., -3).
Logic.
The double negation law.
Marla is a student in this discrete mathematics class.
An n-tuple (p1, ..., pn) of Boolean values.
We will not go swimming.
We will be home by sunset.
C(x) represents 'x has visited Canada'.
A propositional statement that is always false.
q → p.
To convey meaning from one language to another.
If 'P implies Q' and 'P' is true, then 'Q' must also be true.
The proposition 'p if and only if q.'
∃xP(x).
Because it contains a predicate and a quantifier that require more than just logical connectives.
¬q → ¬p.
When P(x) is true for at least one x in the domain.
It indicates that a statement is true for all elements in a domain.
The hypothesis (or antecedent or premise).
In propositional logic, implication does not have to represent rational causality.
∃ x ∀ y (¬P(x,y) ∨ ∀ z ¬R(x,y,z)).
If 'P implies Q' and 'Q implies R', then 'P implies R'.
'If it rains, then the ground will be wet.'
If 'P implies Q' is true and 'Q' is false, then 'P' must also be false.
George Boole.
A statement that is assumed to be true.
For every person y, there exists a person x such that x relies upon y.
Conditional Statement.
1. Everyone in this discrete mathematics class has taken a course in computer science. 2. Marla is a student in this class.
Rules of Inference.
The identity law for F.
Since we are not going swimming, we will take a canoe trip.
Slides by Jan Stelovsky.
If Q is a necessary condition for P, then P cannot be true unless Q is true.
It never leads from correct statements to an incorrect conclusion.
It transforms a proposition into its logical negation.
Forms of argument that can lead from true statements to an incorrect conclusion.
It illustrates the use of a predicate and a quantifier.
q.
p ↔ q is equivalent to (p → q) ∧ (q → p).
True.
True.
It means 'There is an x greater than zero such that P(x).'
A statement that two logical expressions have the same truth value in every possible scenario.
Quantifiers that are placed within the scope of other quantifiers.
(p ∧ (p → q)) → q.
It is equivalent to 'not (not p or not q)'.
Disjunction.
A quantifier that indicates 'for all' values of x.
(¬ p) ∧ q.
As a propositional function P(·) from subjects to propositions.
It combines one or more operand expressions into a larger expression.
If 'P implies Q' and 'Q implies R' are both true, then 'P implies R' is also true.
For every person x and for every person y, x relies upon y.
'The cat is sleeping'.
A logical operation that results in true if exactly one of the propositions is true.
They operate on propositions or their truth values instead of on numbers.
It allows for a wider range of results by including synonyms or related terms.
It implies that p is true, or q is true, or both are true.
Because it includes the possibility that both p and q are true.
To verify the correctness of a program.
We will be home by sunset.
To convey the same meaning in another language.
If the ground is not wet, then it is not raining.
The cat is on the mat.
¬(p ∨ q) and ¬p ∧ ¬q.
Functions that return true or false based on the input values.
For all real numbers x and for all real numbers y, there exists a real number z such that x + y = z.
'The dog'.
p ∨ ¬p ("Today the sun will shine or today the sun will not shine.")
p, p → q, therefore q.
The contrapositive is the inverse of the converse.
De Morgan’s laws for quantifiers.
To determine the truth values of compound propositions.
No, it does not imply that p and q are true.
Symbols or words used to connect search terms in a logical manner.
It reverses the truth value of the quantified statement.
It is equivalent to 'not (not p and not q)'.
¬ (p ∨ (¬ p ∧ q))
'not p or q'.
The study of reasoning.
∃ x ∀ y (¬P(x,y) ∨ ∀ z ¬R(x,y,z)).
If P(x) is false for at least one value of x in the domain.
To ensure that conclusions drawn from premises are logically valid.
An operator that takes two operands (e.g., 3 + 4).
The relationship among statements, not the content of any particular statement.
¬ p ∧ ¬ q.
To build complicated compound propositions involving propositional variables.
Statements that have the same truth value in every possible scenario.
A compound proposition that is true no matter the truth values of its atomic propositions.
A function that returns true or false based on the input values.
Premise 1, Premise 2, Conclusion.
The Foundations: Logic and Proofs.
True, because for any x and y, you can find a z (specifically z = x + y).
¬(p ∨ q) is equivalent to ¬p ∧ ¬q.
Yes, it is a proposition because it can be classified as true.
A branch of logic that deals with propositions and their relationships.
Algorithm design, Machine Learning/Deep Learning on Computer vision, and NLP.
No, it is not a proposition because it contains a variable and cannot be definitively classified as true or false.
Because it affects the meaning, unless all are universal or all are existential quantifiers.
An element x in the domain that makes P(x) false.
A statement that follows from p being true.
The order does not affect the meaning.
Nested Quantifiers.
A logical operation that results in true if at least one of the propositions is true.
It excludes results that contain the specified term.
1. It is not sunny this afternoon and it is colder than yesterday. 2. We will go swimming only if it is sunny. 3. If we do not go swimming, then we will take a canoe trip. 4. If we take a canoe trip, then we will be home by sunset.
It is the statement 'It is not the case that p.'
By successively applying the rules for negating statements involving a single quantifier.
All, some, many, none, few, etc.
Propositional calculus.
They yield the same truth values, proving they are logically equivalent.
It indicates that there exists at least one element in a domain for which the statement is true.
They represent the same truth values under all circumstances.
p ∧ ¬p ("Today is Wednesday and today is not Wednesday.")
A property that the subject of the statement can have.
AND, OR, NOT.
A quantifier that indicates 'there exists' a value of x such that ...
He is the person behind the 'Boolean' name and discussed it in his 1854 book 'The Laws of Thought.'
A logical operation that results in true only if both propositions are true.
'not p or not q'.
It is denoted by p ∨ q.
Precise meaning.
Modus Ponens.
If 'P implies Q' and 'P' is true, then 'Q' must also be true.
To represent the truth values of propositions and their logical operations.
Expressing conditions.
If 'P implies Q' is true and 'P' is true, then 'Q' must also be true.
By using parentheses.
(p ∨ ¬q) → (p ∧ q).
False, because no single z can satisfy x + y = z for all x and y.
Yes, there are many ways to express conditional statements.
It is not equivalent to ¬ p ∧ q.
The binary disjunction operator is '∨' (OR).
Both have a prioritization in operations, with negation in logic similar to multiplication/division in arithmetic.
Yes, it is a proposition because it can be classified as true.
It represents 'p or q'.
Introduction to Proofs.
A statement that expresses a logical relationship between two propositions, typically in the form 'if P, then Q'.
Proof Methods and Strategy.
F (False).
Marla has taken a course in computer science.
Using a truth table.
