Math Problem Statement

( R , = ) is this poset a lattice?

Solution

To determine whether the poset (R,)(\mathbb{R}, \leq) is a lattice, we must understand the definition of a lattice and check if (R,)(\mathbb{R}, \leq) satisfies it.

Lattice Definition

A poset (P,)(P, \leq) is a lattice if for every pair of elements a,bPa, b \in P, there exist:

  1. A least upper bound (also known as the supremum or join), denoted by aba \vee b, which is the smallest element in PP greater than or equal to both aa and bb.
  2. A greatest lower bound (also known as the infimum or meet), denoted by aba \wedge b, which is the largest element in PP less than or equal to both aa and bb.

Poset (R,)(\mathbb{R}, \leq)

In the poset (R,)(\mathbb{R}, \leq):

  • The set R\mathbb{R} represents the set of all real numbers.
  • The relation \leq is the usual order on the real numbers.

Analyzing the Poset

For any two real numbers a,bRa, b \in \mathbb{R}:

  • The least upper bound (supremum) of aa and bb exists and is given by max(a,b)\max(a, b).
  • The greatest lower bound (infimum) of aa and bb exists and is given by min(a,b)\min(a, b).

Since both the supremum and infimum exist for every pair of elements in R\mathbb{R}, the poset (R,)(\mathbb{R}, \leq) is indeed a lattice.

Conclusion

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Math Problem Analysis

Mathematical Concepts

Posets
Lattices

Formulas

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Theorems

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Suitable Grade Level

Advanced Mathematics