01204211/activity1 logic1

In-class activities

1. Let propositions $P$ = "you go to see the movie Harry Potter," $Q$ = "the movie Harry Potter is good," and $R$ = "you have a good time.". Express each sentence below as a propositional form using variables $P,Q,$ and $R$.

1.1 You do not go to see the movie Harry Potter.

1.2 If Harry Potter is good and you go to see it, you will have a good time.

1.3 You can have a good time, even if you do not go to see the movie Harry Potter.

1.4 If you go to see the movie Harry Potter and do not have a good time, the movie Harry Potter must be bad.

2. For each of these sentences, define appropriate propositional variables representing each proposition inside the statement and translate the statement into a propositional form.

2.1 It is raining or it is very hot.

2.2 If you like Thai food, you will enjoy the trip to the Night Market.

2.3 The only way you can finish a marathon is that you practice a lot and have strong will to fight.

2.4 You either learn to understand the customer or you fail to make a good product.

Quantifiers

3. Consider the universe to be "everything." For each of these statements, define appropriate predicates can rewrite the statement using the defined predicates and quantifiers. (Some predicate may have more than one variables)

3.1 Every human must die.

3.2 Some animal eats other animals.

3.3 If a student works hard, that student will be successful.

For questions 3.4 and 3.5, consider the universe to be a set of all people.

3.4 Everyone has someone that care about him or her.

3.5 There is someone that everyone cares about.

For questions 3.6 and 3.7, consider the universe to be a set of all companies.

3.6 When the economy is good, any companies can make good profits.

3.7 When the economy is bad, only companies that can adapt survive.

4. For each quantified proposition you answer in question 3, find its negation and translate the negation back to English.

5. It seems that universal quantifiers are stronger than existential ones. Is it true that for any set $A$ and predicate $P(x)$,

$(\forall x\in A,P(x))\Rightarrow (\exists x\in A,P(x))$?

6. Use a truth table to show that $P\Rightarrow Q$ is equivalent to $\neg P\vee Q$. Use this equivalence to prove that $P\Rightarrow Q\equiv \neg Q\Rightarrow \neg P$.

Homework

Inference rules

This part should be attempted after the instructor has discussed exhaustive proof technique and inference rules.

7. Use a truth table to prove Hypothetical syllogism. That is show that the conclusion $P\Rightarrow R$ logically follows from hypotheses $P\Rightarrow Q$ and $Q\Rightarrow R$.

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