Solution to Question #221673 Say for each of the posets represented by the given Hasse-diagram whether the … - Sikademy
Author Image

Archangel Macsika

Question #221673 Say for each of the posets represented by the given Hasse-diagram whether the poset is i) a lattice ii) a complemented lattice iii) a Boolean algebra Give reasons for your answers

The Answer to the Question
is below this banner.

Can't find a solution anywhere?


Get the Answers Now!

You will get a detailed answer to your question or assignment in the shortest time possible.

Here's the Solution to this Question


Here Hasse diagram is missing, so we will define the given terms:

(a) A poset is short for partially ordered set which is a set whose elements are ordered but not all pairs of elements are required to comparable in the order.

(i) A lattice is a poset (𝑋, 𝑅) with two properties: • 𝑋 has an upper bound 1 and a lower bound 0; • for any two elements 𝑥, 𝑦 ∈ 𝑋, there is a least upper bound and a greatest lower bound of a set {𝑥, 𝑦}. 

(ii) A complemented lattice is an algebraic structure \left(L, \wedge, v, 0,1,^{\prime}\right) such that (L, \wedge, v, 0,1) is a bounded lattice and for each element x \in L , the element x^{\prime} \in L is a complement of x, meaning that it satisfies

1. x \wedge x^{\prime}=0

2. x \vee x^{\prime}=1

(iii) A Boolean algebra (BA) is a set A together with binary operations + and \cdot and a unary operation -, and elements 0,1 of A such that the following laws hold: commutative and associative laws for addition and multiplication, distributive laws both for multiplication over addition and for addition over multiplication, and the following special laws:

\begin{array}{r} (x+(x \cdot y)=x \\ (x \cdot(x+y)=x \\ x+(-x)=1 \\ x \cdot(-x)=0 \end{array}

Related Answers

Was this answer helpful?

Join our Community to stay in the know

Get updates for similar and other helpful Answers

Question ID: mtid-5-stid-8-sqid-2749-qpid-1219