Description

Discrete Mathematics
Karachi, Pakistan
Recap of the Great Rules

Question: State which rule of inference is the basis of the following argument: “It is below freezing now. Therefore, it is either below freezing or raining now.”

Solution: Solution: Let p be the proposition “It is below freezing now” and q the proposition “It is raining now.” Then this argument is of the form
p

p ∨ q
This is an argument that uses the addition rule.
State which rule of inference is the basis of the following argument: “It is below freezing and raining now. Therefore, it is below freezing now.”
Solution: Solution: Let p be the proposition “It is below freezing now,” and let q be the proposition “It is raining now.” This argument is of the form
p ∧ q

p
This argument uses the simplification rule.
Solution: Let

Solution:
Let
1 ¬p ∧ q Premise
2 ¬p Simplification using (1)
3 s =⇒ p Premise
4 ¬s Modus tollens using (2) and (3)
5 ¬s =⇒ r Premise
6 r Modus ponens using (4) and (5)
7 r =⇒ t Premise
8 t Modus ponens using (6) and (7)

DIY TIME
DIY
The following questions are for you to do yourself rather then me explaining. I’m here to help and support you, I am here for if you wanna talk about it.
Question 1: What is true?
The following questions are for you to do yourself. I’m here to help and support you For each of these arguments determine whether the argument is valid or invalid and explain why.
1 a) All students in this class understand logic. Affan is a student in this class. Therefore, Affan understands logic.
2 b) Every computer science major takes discrete mathematics. Khubaib is taking discrete mathematics. Therefore, Khubaib is a computer science major.
3 c) All TAs are lazy. Some students are not TAs. Therefore, those students are not lazy.
4 d) Everyone who eats Andey wala burger every day wears a kameez. Mujtaba does not wear a kameez. Therefore, Mujtaba does not eat Andey wala burger every day.
Question 2 : Is this Real life?
The following questions are for you to do yourself rather then me explaining. I’m here to help and support you.
Question 3: Or is this Fanta-sea?
The following questions are for you to do yourself rather then me explaining. I’m here to help and support you.
1 What is wrong with this argument? Let H(x) be “x can’t wait for the new Demon Slayer season.” Given the premise ∃xH(x), we conclude that H(Ifrah). Therefore,
“Ifrah can’t wait for the new Demon Slayer season.”
2 What is wrong with this argument? Let S(x,y) be “x is smarter than y.” Given the premise ∃sS(s,Kanye West), it follows that S(Kanye West,Kanye West). Then by existential generalization, it follows that ∃xS(x,x), so that Kanye West is smarter than himself.
Question 4: Compound are scary
The following questions are for you to do yourself. I’m here to help and support you, Send me the solution for each in our MS Teams channel.
Use resolution to show that the compound proposition
(p ∨ q) ∧ (¬p ∨ q) ∧ (p ∨¬q) ∧ (¬p ∨¬q) is not satisfiable.
Question 5: Proof.exe is not responding
The following questions are for you to do yourself. I’m here to help and support you, Send me the solution for each in our MS Teams channel.
Identify the error or errors in this argument that supposedly shows that if
∃xP(x) ∧∃xQ(x) is true then ∃x(P(x) ∧ Q(x)) is true
1 1. ∃xP(x) ∨∃xQ(x) Premise
2 2. ∃xP(x) Simplification from (1)
3 3.P(c) Existential instantiation from (2)
4 4. ∃xQ(x) Simplification from (1)
5 5. Q(c) Existential instantiation from (4)
6 6. P(c) ∧ Q(c) Conjunction from (3) and (5)
7 7. ∃x(P(x) ∧ Q(x)) Existential generalization

Conclusion
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