   Chapter 8.5, Problem 48ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Use the algorithm given in the text to find a topological sorting for the “subset” relation on P ( { a , b , c , d } ) .

To determine

To find the topological sorting for the subset “subset” relation on the given set.

Explanation

Given information:

The given subset relation on the power set P={a,b,c,d}.

Calculation:

The following is the algorithm given using which the topological sorting is to be found out for the given relation-

The relation is partial order on the sets. So now using the algorithm on the set A

1. Pick any minimal element x in A.
2. Set A'=A{x}.
3. Repeat the following steps ac while A'=ϕ

a) Pick any element in A'.

b) Define xy.

c) set A':=A{y}

So now let us follow the above algorithm for the given power set A

{ϕ,{a},{b},{c},{d},{a,b},{a,c},{b,c},{a,d},{b,d},{b,c},{c,d},{a,b,d},{a,c,d},{b,c,d},{a,b,c,d}}

The set has the minimal element ϕ, so removing it,

The set will be A'=Aϕ. Now the remaining minimal elements will be a,b,c.

So till now the topological sorting order will be ϕ{a}, ϕ{b}, ϕ{c}.

Also, for the second set of minimal elements as {a}{b}{c}.

Further pick the next level of elements {a}{a,b},{a}{a,c}, {b}{a,b},{b}{a,c}, {c}{a,b},{c}{a,c}.

So till now the topological order sorting is ϕ{a}{b}{c}{a,b}{a,c}{b,c}.

Now continuing in the same way, the next level of ordering {a,b}{a,b,c},{b,c}{a,b,c},{a,c}{a,b,c}

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