01204211/activity1 logic1

This is part of 01204211-58.

In-class activities 1

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 You can finish a marathon, only if you practice a lot and have strong will to fight.

(Edited from: 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.

REMARKS: You may need to use a few logical equivalences to complete question 4,
for example, $P\Rightarrow Q\equiv \neg P\vee Q$.
I'll add more information on that soon.

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 1

Due: 2 Sep 2015

Quantifiers

1. (Source: wikibooks) The following predicates are defined:

friend(x) is "x is a friend of mine"

wealthy(x) is "x is wealthy"

clever(x) is "x is clever"

boring(x) is "x is boring"

With these predicates, you can write "John is clever" as clever(John).

Write each of the following propositions using predicate notation:

1.1 Jimmy is a friend of mine.

1.2 Sue is wealthy and clever.

1.3 Jane is wealthy but not clever.

1.4 Both Mark and Elaine are friends of mine.

1.5 If Peter is a friend of mine, then he is not boring.

1.6 If Jimmy is wealthy and not boring, then he is a friend of mine.

Questions 2 and 3 are from wikibooks.

2. (Source: [1]) Using the same predicates you defined in question 1, symbolize each of the following.

(a) Some of my friends are clever.

(b) All clever people are boring.

(c) None of my friends is wealthy.

(d) Some of my wealthy friends are clever.

(e) All my clever friends are boring.

(f) All clever people are either boring or wealthy.

3. (Source: [2]) Define suitable predicates, and hence symbolize:

(a) All pop-stars are overpaid.

(b) Some RAF pilots are women.

(c) No students own a Rolls-Royce.

(d) Some doctors cannot write legibly.