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PHL246H1 (19)

Homework assignment 4(1).pdf

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University of Toronto St. George
Franz Huber

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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