Question

Find the tasks needed to build a house, by specifying their order.

Short answer

The task of, to build a house is

\(P = \{ a,b,c,d,e,f,g\} {\rm{. }}\)Then,

\(R = \left\{ {\begin{array}{*{20}{l}}{(a,a),(a,g),(a,d),(a,e),(a,f),}\\{(b,b),(b,g),(b,d),(b,e),(b,f),}\\{(c,c),(c,g),(c,d),(c,e),(c,f)}\\{(g,d),(g,e),(g,f),(g,g),(d,d)}\\{(e,e),(f,f),}\end{array}} \right\}\)

Step-by-step solution

Given data

The Hasse diagram representing these tasks is shown in the figure.

Completion

Interior fixtures

Exterior fixtures

Carpeting

Flooring lnterior painting

Exterior painting

Plumbing wall board

Winning

Exterior siding Roof

Framing

Foundation.

Concept used of poset

A relation\(R\)is a poset if and only if,\((x,x)\)is in\({\rm{R}}\)for all x (reflexivity)

\((x,y)\)and\((y,x)\)in R implies\(x = y\)(anti-symmetry),\((x,y)\)and\((y,z)\)in R implies\((x,z)\)is in\({\rm{R}}\)(transitivity).

Find the tasks needed

We consider the provided statement and figure from textbook to schedule the

tasks that needed to build a house in order.

For determine one possible order of task do a topological sort. There are many compatible

total orders; one possible order is shown as below; Foundation, Framing Roof, Exterior

Siding, Wiring, Plumbing, Flooring, Wallboard, Exterior Painting, Interior Painting,

Carpeting, Interior Fixtures, Exterior Fixtures, Completion.

The above is a Hasse diagram of a poset \((P,R)\), where

\(P = \{ a,b,c,d,e,f,g\} {\rm{. }}\)

Then,\(R = \left\{ {\begin{array}{*{20}{l}}{(a,a),(a,g),(a,d),(a,e),(a,f)}\\{(b,b),(b,g),(b,d),(b,e),(b,f)}\\{(c,c),(c,g),(c,d),(c,e),(c,f)}\\{(g,d),(g,e),(g,f),(g,g),(d,d)}\\{(e,e),(f,f),}\end{array}} \right\}\)

Hence this is required solution.