   Chapter 1.7, Problem 30E

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# Suppose that f is an onto mapping from A to B . Prove that if { B λ } , λ ∈ ℒL, is a partition of B , then {   f − 1   ( B λ ) } , λ ∈ ℒL, is a partition of A .

To determine

To prove: If {Bλ},λL is a partition of B, then {f1(Bλ)},λL is a partition of A.

Explanation

Given Information:

f is an onto mapping from A to B.

Formula Used:

Let {Aλ},λL be a collection of subsets of the nonempty set A. Then {Aλ},λL is a partition of A if all these conditions are satisfied:

Each Aλ is nonempty.

A=λL{Aλ}.

If AαAβ then Aα=Aβ.

Proof:

f is an onto mapping from A to B.

R is an equivalence relation on the nonempty set A.

{Bλ},λL is a partition of B.

To show that {f1(Bλ)},λL is a partition of A:

Let bB

By using f is an onto mapping from A to B,

There exists an element aA such that f(a)=b.

Thus f(a)Bλ for some λL

Hence af1(Bλ)

Thus f1(Bλ) ……… (1)

If f1(Bα)f1(Bβ)

Thus there exists cf1(Bα)f1(Bβ)

cf1(Bα) and cf1(Bβ)

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