Question

For the given poset \((\{ \{ 1\} ,\{ 2\} ,\{ 4\} ,\{ 1,2\} ,\{ 1,4\} ,\{ 2,4\} ,\{ 3,4\} ,\{ 1,3,4\} ,\{ 2,3,4\} 1, \subseteq )\) find the greatest lower bound of \(\{ 1,3,4\} ,\{ 2,3,4\} \).

Short answer

The greatest lower bound of \(\{ 1,3,4\} ,\{ 2,3,4\} \) is \(\{ 3,4\} \).

Step-by-step solution

Given data

Given data is \((\{ \{ 1\} ,\{ 2\} ,\{ 4\} ,\{ 1,2\} ,\{ 1,4\} ,\{ 2,4\} ,\{ 3,4\} ,\{ 1,3,4\} ,\{ 2,3,4\} 1, \subseteq )\).

Concept used of greatest lower bound

Let\(A\)be an ordered set and\(X\)a subset of\(A\). An element\(b\)is called a lower bound for the set\(X\)if every element in\(X\)is greater than or equal to\(b\). If such a lower bound exists, the set\(X\)is called bounded below.

Find greatest lower bound

We start by drawing the Hasse diagram corresponding to the poset \((\{ \{ 1\} ,\{ 2\} ,\{ 4\} ,\{ 1,2\} ,\{ 1,4\} ,\{ 2,4\} ,\{ 3,4\} ,\{ 1,3,4\} ,\{ 2,3,4\} 1, \subseteq )\)

The element \(y\) is called the greatest lower bound of \(A\) if \(y\) is a lower bound of \(A\) and \(z \le y\) whenever \(z\) is a lower bound of \(A\).

Clearly in this Hasse diagram the greatest lower bound of \(\{ 1,3,4\} ,\{ 2,3,4\} \) is \(\{ 3,4\} \) .