## Questions and Answers

What is the definition of an existential quantifier?

What does the universal quantifier express?

How is an existential quantifier symbolized?

What does it mean for a predicate to be satisfiable?

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Which quantifier represents 'for every' in predicate logic?

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In predicate logic, what does $P(x) \lor Q(x)$ represent?

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What does a valid predicate imply?

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How can a predicate with free variables be assigned a truth value?

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'There exists an x such that P(x) is true' can be symbolized as:

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What does 'for every x and for every y, x+y > 10' represent?

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## Study Notes

### Introduction to Logic

- Logic is the study of reasoning and the validity of arguments, concerned with the truth of statements (propositions) and the nature of truth.
- Formal logic is concerned with the form of arguments and the principles of valid inference.

### Propositional Logic

- Propositional logic is the study of propositions, which are declarative statements that can result in either true or false.
- Propositions may be combined with other propositions using logical connectives to form compound propositions.
- Propositional logic can be used to encode simple arguments expressed in natural language and to determine their validity.

### Propositional Connectives and Truth Tables

- Five propositional connectives, in descending order of operator precedence:
- Truth tables are used to determine whether a propositional statement is true or false for all possible instances of the variable.

### Conjunction and Disjunction

- The conjunction p ^ q is true exactly when p is true and q is true.
- The conjunction p ^ q is false when p is false or q is false or both are false.

### Implication and Equivalence

- The implication p=>q is true in every case except when p is true and q is false.
- The implication p=>q is true if and only if we can prove q by assuming p.

### Negations, Tautologies, and Contradictions

- A negation of a proposition is true when the original proposition is false, and false when the original proposition is true.
- Tautologies are propositions that evaluate to true in every combination of their propositional variables.
- Contradictions are propositions that evaluate to false in every combination of their propositional variables.

### Predicate Logic

- Predicate logic is a richer system than propositional logic, allowing complex facts about the world to be represented.
- Predicate calculus consists of predicates, variables, constants, and quantifiers.

### Predicates

- A predicate is a statement that contains variables (predicate variables), which may be true or false depending on the values of these variables.
- The domain of a predicate variable is the collection of all possible values that the variable may take.

### Quantifiers: Universal and Existential

- The universal quantifier states that a predicate is true for all possible values in the universe of discourse.
- The existential quantifier states that there exists a value in the universe of discourse for which a predicate is true.

### Satisfaction and Validity

- A predicate with free variables or 'spaces' is neither true nor false until values are chosen for these variables or the spaces are filled.
- A predicate that is true for all choices of values is said to be valid.
- A predicate that is true for some, but not necessarily all, choices of values is said to be satisfiable.

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## Description

Test your knowledge on propositional logic including conjunction, disjunction, implication, equivalence, negations, tautology, and contradictions. Explore predicate logic with quantifiers like universal and existential, satisfaction, and validity.