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, this sentence is not a statement because it is an opinion. A correction might be, "The Caribbean's weather is typically around 80 degrees."
Consider the following: ¬(¬P) ≡ P. What does this mean?
This means that ¬(¬P) is logically equivalent to P, such that both expressions will have the same truth value in every scenario.
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 ______.”
1) Fill in the blank to make a statement.
2) Provide a quantifier for the statement to make it true.
1) "Taco Bell sells burritos that cost a dollar or more."
2) For all burritos Taco Bell sells, they cost a dollar or more.
Create a circuit for the following expression: x∧(y→(¬z))
Jeopardy doesn't allow pictures unless you pay. I'll attach the circuit diagram separately! :)
What is a Tautology? Converse statement?
A tautology is a statement that is always true.
A converse is the reverse order of an implication. Meaning, if the implication is (P → Q), then the converse is (Q → P).
P = “Sam goes to school”, Q = “Sam studies”
1) Write an implication(P→Q) for the above parts.
2) “If Sam studies, then he goes to school”. What kind of sentence is this?
1) "If Sam goes to school, then he studies."
2) Converse of the implication.
“X represents a number that is bigger than 2x or smaller than ½x.”
1) What kind of sentence is this?
2) Is it true? Provide an example.
3) What is this example from part b called?
1) This sentence is a predicate.
2) No, the sentence is not true.
Consider the following:
Let x=2. We have:
2 ≯ 2(2), 2 ≮ ½(2).
This simplifies to 2≯ 4, 2 ≮ 1. Thus, we have found that the sentence “X represents a number that is bigger than 2x or smaller than ½x" is false.
3) We call the example from part (b) a counterexample.
What is the value of P → P? Is there anything special about this expression?
It is true. Moreover, it is a tautology!
Given the statement “If a car has a good gas mileage, then you can save money”, find the:
1) Converse
2) Contrapositive
3) 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 gas mileage, then you can save money."
Create a circuit diagram logically equivalent to the diagram Professor Mock draws!
Jeopardy doesn't allow pictures unless you pay. I'll attach the circuit diagram separately! :)
Consider the following pseudocode:
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if team1_score > 3
print(“Team1 wins”)
else
print(“Team1 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), ¬P∨¬Q is equivalent to ¬(P∧Q), ¬P∧¬Q is equivalent to ¬(P∨Q), etc.
Is (P∧Q)→R ≡ P∧(Q→R)?
Not logically equivalent.
Since the boolean operators between the boolean variables are not all the same, the associative law does not apply.
OR
Draw the truth table and find that truth values in every column and row do not match for (P∧Q)→R and P∧(Q→R).
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?
This sentence is a predicate.
No, this is not true. Consider the following:
Let x = 4, we have:
4 > 3, but 4 ≯ 5
Thus, we have that the sentence “If x > 3 then x > 5” is false.
Provide a truth table for ¬P∧(Q⊕R) ≡ (P⊙¬Q)→(¬P⊻P)
Jeopardy doesn't allow pictures unless you pay. I'll attach the circuit diagram separately! :)
Simplify the following expression and design a circuit diagram for it:
( ¬x∨¬y)∧(¬x∨¬z)
Jeopardy doesn't allow pictures unless you pay. I'll attach the circuit diagram separately! :)
Is (P∨Q) ≡ (P⊼P)⊼(Q⊼Q)?
Yes, they're logically equivalent.
Draw a truth table to support this answer. Follow the truth table with the statement, "Since all truth values in every column and row match, we know (P∨Q) is logically equivalent to (P⊼P)⊼(Q⊼Q)."
OR
Use the correct Laws to show the equivalence.
Using disjunctive normal form, design a minimalized circuit diagram to represent the chart Professor Mock puts on the board!
We did this question in class on Friday! :)