Definition 2.3.2 in our book defines the power set of a set S, denoted by P(S),...

Definition 2.3.2 in our book defines the power set of a set S, denoted by P(S), as the set of all subsets of S, that is P(S) = {A : A ⊆ S}. For example, P({1, 2}) = {∅, {1}, {2}, {1, 2}}, and P(∅) = {∅}. Consider the relation ⊆ on the power set P(Z), i.e. the is-a-subset-of relation defined on sets of integers. In other words, the objects we compare are sets of integers. (a) Prove or disprove: ⊆ is reflexive. (b) Prove or disprove: ⊆ is irreflexive. (c) Prove or disprove: ⊆ is symmetric. (d) Prove or disprove: ⊆ is antisymmetric. (e) Prove or disprove: ⊆ is transitive. (f) Is ⊆ on Z an equivalence relation? Is it a partial order? (g) Which other relation satisfies exactly the same properties?

Expert Solution

Colby Messinger answered 2 years ago

Let S = {a, b, c, d} and P(S) its power set. Define the minus binary...

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Let S ⊆ R be a nonempty compact set and p ∈ R. Prove that there...

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Let Prob be a probability function on the power set P(Ω) of a sample space Ω....

Let Prob be a probability function on the power set P(Ω) of a sample space Ω. Let B be a set such that Prob(B) > 0. For any set A, define G(A|B) = Prob(A ∩ B) Prob(B) . Prove that for fixed B and as as a function of A, G(·|B) is also a probability function, meaning that G(·|B) satisfies the axioms of probability..

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