Column 1
“The Caribbean has warm weather.”
Is the above sentence a statement?
If so, why? If not, provide a correction to make it a statement.
No.
The Caribbean has warmer weather than the Arctic.
Consider the following: ¬(¬P) ≡ P. What kind of equivalence is this?
Tautology
What do we call P ∧ (¬ P)
Contradiction
Name all of the boolean operators.
Conjunction, Disjunction, Negation, Nand, Nor, Xor, Xnor
Consider the following: “Taco Bell sells burritos that ______.”
Fill in the blank to make a statement.
Provide a quantifier for the statement to make it true.
Taco Bell sells burritos that cost a dollar or more.
For all burritos Taco Bell sells, they cost a dollar or more.
Create a circuit for the following expression: x∧(y→(¬z))
Reminder: x∧(y→(¬z)) is logically equivalent to
x∧(¬z ∨¬y)
Professor Mock will check your circuit! (sheet)
What is a Tautology? Converse statement?
A statement that is always true.
A converse is the reverse order of an implication. If the implication is P → Q, then the converse is Q → P
P = “Sam goes to school”, Q = “Sam studies”
Write an implication for the above parts in the order of P, then Q.
“If Sam studies, then he goes to school”. What kind of sentence is this?
If Sam goes to school, then he studies.
Converse of the implication
“x represents a number that is bigger than 2x or smaller than ½x.”
What kind of sentence is this?
Is it true? Provide an example.
What is this example called?
Predicate
No. x=2 → 2 < 2(2) = 4 and 2 > ½(2) = 1
Contradiction
What is the value of P → P? Is there anything special about this expression?
It is true. It is a tautology.
Given the statement “If a car has a good gas mileage, then you can save money”, find the:
Converse
Contrapositive
Negation
If you can save money, then a car has good gas mileage.
If you can't save money, then a car doesn't have good gas mileage.
It is not the case that if a car has good mileage, then you can't save money.
Create a circuit diagram logically equivalent to the diagram Professor Mock draws! (16)
Professor Mock will check your diagram! (16)
Consider the following pseudocode:
—————————————————
if team1_score >3
print("Team 1 wins")
else
print("Team 1 loses")
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Edit this program such that it is written in terms of the converse.
if Team1 wins
print(“team1_score > 3”)
else
print(“team1_score <= 3”)
Provide an example of a boolean expression, and then write its equivalent form using DeMorgan’s Law.
P∨¬Q is equivalent to ¬(¬P∧Q)
Is (P∧Q)→R ≡ P∧(Q→R)?
Not logically equivalent. (7)
Picture this: you’re in your Discrete Structures class and Professor Mock puts the following on the board: “If x > 3 then x > 5.”
What kind of sentence is this?
Is it true? Provide an example.
What is this example called?
Predicate
No. Let x = 4. 4 > 3 = T and 3 > 5 = F. T → F is false.
Contradiction
Provide a truth table for ¬P∧(Q⊕R) ≡ (P⊙¬Q)→(¬P⊻P)
Professor Mock will check your truth table! (4)
Simplify the following expression and design a circuit diagram for it: (¬x∨¬y)∧(¬x∨¬z)
¬(x⋀(y∨z))
Professor Mock will check your diagram! (17)
Is (P∨Q) ≡ (P⊼P)⊼(Q⊼Q)?
Yes, they're logically equivalent.
Using disjunctive normal form, design a minimalized circuit diagram to represent the chart Professor Mock puts on the board! (20)
Professor Mock will check your diagram! (20)