{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

discrete-structures

# 6 if laura is not at the office then john and joan

This preview shows pages 27–39. Sign up to view the full content.

6. If Laura is not at the office then John and Joan are both at the office.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.”
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
LOGICAL STRUCTURE 5. Associative properties p (q r) ≡ (p q) r ᴧ ᴧ ᴧ ᴧ p (q r) ≡ (p q) r ᴠ ᴠ ᴠ ᴠ 6. Distributive properties p (q r) ≡ (p q) (p r) ᴧ ᴠ ᴧ ᴠ ᴧ p (q r) ≡ (p q) (p r) ᴠ ᴧ ᴠ ᴧ ᴠ 7. Equivalence of Contrapositive p→q ≡ ¬q→¬p

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 .
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
LOGICAL STRUCTURE To create proposition from a propositional function: Assign values to the variables Quantification types: 1. universal quantification 2. existential quantification

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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) ” ; “for every x P(x) true if P(x) is true for every x ) ( x xP 2200 2200 ) ( x xP 2200
LOGICAL STRUCTURE Examples: Domain for x is the set of real numbers 1. P(x): “x+1>x” 2. P(x): “x>5” 3. Q(x): “x 2 >= x” Note: is false if P(x) is false for at least one x in D. A value x in the domain of discourse that makes P(x) false is called a ) ( x xP 2200 ) ( x xP 2200

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
LOGICAL STRUCTURE Definition The existential quantification of P(x) is the proposition “There exist an element x in the domain of discourse such that P(x) is true.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page27 / 100

6 If Laura is not at the office then John and Joan are both...

This preview shows document pages 27 - 39. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online