discrete-structures

# Q joan is at the office r laura is at the office use

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q: Joan is at the office. r: Laura is at the office. Use logical connectives to express the following sentences: 1. John is not at the office. 2. If Joan and Laura are at the office then John is at the office.

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LOGICAL STRUCTURE 3. If John is at the office then either Joan or Laura is at the office. 4. John, Joan and Laura are all at the office. 5. Joan is not at the office and either John or Laura are at the office. 6. If Laura is not at the office then John and Joan are both at the office.
LOGICAL STRUCTURE B. Translate the following into logical expression. 1. “ You can access the internet from campus only if you are a computer science major or you are not a freshman.” 2. “ You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old.”

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LOGICAL STRUCTURE C. Solve a Crime Four friends have been identified as suspects for an unauthorized access into a computer system. They have made statements to the investigating authorities. Alice said “Carlos did it.” John said “ I did not do it .” Carlos said “ Diana did it.” Diana said “Carlos lied when he said that I did it.” If the authorities also know that exactly one of the four suspects is telling the truth,
LOGICAL STRUCTURE Theorem The following list of logically equivalent properties can be established using truth tables. 1. Indempotent Laws p p ≡ p p p ≡ p 2. Double Negation ¬(¬p) ≡ p 3. De Morgan’s Laws ¬ (p q) ≡ ¬p ¬q

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LOGICAL STRUCTURE 5. Associative properties p (q r) ≡ (p q) r p (q r) ≡ (p q) r 6. Distributive properties p (q r) ≡ (p q) (p r) p (q r) ≡ (p q) (p r) 7. Equivalence of Contrapositive p→q ≡ ¬q→¬p 8. Other useful properties
LOGICAL STRUCTURE QUANTIFIERS Definition Let P(x) be a statement involving the variable x and let D be a set. We call P a propositional function (with respect to D ) if for each x in D , P(x) is a proposition. We call D the domain of discourse of P .

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LOGICAL STRUCTURE P(x) : value of the propositional function P of x . Example: “x>5” these statement is not a proposition because whether it is true or false depends on the value of x. Once a value has been assigned to the variable x, the statement P(x) becomes a proposition and has a truth value.
LOGICAL STRUCTURE In general, a statement involving n variables x 1 , x 2 ,…,x n can be denoted by P(x 1 , x 2 ,…,x n ). A statement of the form P(x 1 , x 2 , …,x n ) is the value of the propositional function P at the n-tuples (x 1 , x 2 ,…,x n ) and P is also called a predicate.

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LOGICAL STRUCTURE To create proposition from a propositional function: Assign values to the variables Quantification types: 1. universal quantification 2. existential quantification
LOGICAL STRUCTURE Definition The universal quantification of P(x) is the proposition “P(x) is true for all values of x in the domain of discourse.” Notation: denotes the universal quantification of P(x) called the universal quantifier read as “for all x P(x)

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• Winter '99
• AverrÃ³is
• Logic, logical structure

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