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# Homework assignment 4(1).pdf

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University of Toronto St. George

Philosophy

PHL246H1

Franz Huber

Fall

Description

Homework assignment 4 (due October 24)
A partition P of an arbitrary non-empty set W is a set of subsets of W, P ⊆ ℘(W), such that any two
members B and C of P are mutually exclusive (have no members in common), B∩C = ∅, and the
members of P are jointly exhaustive (every member of W is contained in some member ofP), ∪P = W.
The symbol ‘∪P’ denotes the union of all members of P. If B, C, D, … are all the members of P, then ∪P =
B∪C∪D∪… Formally it is defined as: ∪P = {x: x ∈ S for some S ∈ P}. ∩P is defined similarly.
The set {{Parisa, Seya}, {Franz}} is a partition of T, and so are {{Parisa}, {Seya}, {Franz}} and {∅, T}.
Consider the set S of all students enrolled in PHL 246. Each student has a certain age. Show that the
following set is a partition of S by establishing that its members are mutually exclusive and jointly
exhaustive:
{{x ∈ S: x is younger than 20 years}, {x ∈ S: x is at least 20 years, but less than 30 years}, {x ∈ S: x is older
than 30 years}} (2 points)
An algebra A over an arbitrary non-empty set W is a set of subsets of W, A ⊆ ℘(W), such that:
(i) W is a member of A, W ∈ A;
(ii) if B is a member of A, then the complement of B with respect to W is also a member ofA; i.e. if B ∈ A,
then W\B ∈ A;
(iii) if B and C are members of A, then t

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