# Assignment #5 + Solution Winter 2009

CS 245 Winter 2009

Assignment 5

Due: Thu 12 Mar 2009 10am in the CS245 Drop Boxes

35 marks

SOLUTION SET

There may be multiple correct answers to some of these questions.

1. (9 marks) Formalize the following sentences in set theory. Do not use types, quantiﬁers,

or set comprehension. Use only the following sets and relations in your formulas:

•Dwellings – the set of dwelling

•People – the set of people

•Houses – the set of houses

•Students – the set of students

•owns :People ↔Dwellings

The relationship between people and the dwellings that they own.

•rents :People ↔Dwellings

The relationship between people and the dwellings that they rent.

where Students ⊆People,Houses ⊆Dwellings.

1

Sentences to formalize:

(a) All houses that are rented are owned.

(3 marks)

(ran rents)∩Houses ⊆ran owns

(b) Not every student owns a house.

(3 marks)

¬(Students ⊆dom (owns ⊲Houses))

or

dom (Students ⊳(owns ⊲Houses)) ⊂Students

(c) Students who rent houses do not own any dwelling.

(3 marks)

(dom (Students ⊳(rents ⊲Houses))) ⊳owns =∅

or

(dom (Students ⊳(rents ⊲Houses))) ∩(dom owns) = ∅

or

owns(|(dom (Students ⊳(rents ⊲Houses))) |) = ∅

## Document Summary

Due: thu 12 mar 2009 10am in the cs245 drop boxes. There may be multiple correct answers to some of these questions: (9 marks) formalize the following sentences in set theory. Do not use types, quanti ers, or set comprehension. The relationship between people and the dwellings that they own: rents : people dwellings. The relationship between people and the dwellings that they rent. where students people, houses dwellings. Sentences to formalize: (a) all houses that are rented are owned. (3 marks) (ran rents) houses ran owns (b) not every student owns a house. (3 marks) You may use natural deduction or trans- formational proof. In both types of proofs, you may use the axioms and derived laws of set theory. If you use natural deduction, do not use any logical laws from transfor- mational proof. In transformational proof, you may use the logical laws of predicate and propositional logic.