Consider the partial order ≤on the set X of positive integers given by "is a divisor of." Let a

step 1 Okay, so we want to show that if A divides C, B divide C, then this implies that A B divides C D

Consider the partial order ≤on the set X of positive...

Consider the partial order $\leq$ on the set $X$ of positive integers given by "is a divisor of." Let $a$ and $b$ be two integers. Let $c$ be the largest integer such that $c \leq a$ and $c \leq b$, and let $d$ be the smallest integer such that $a \leq d$ and $b \leq d .$ What are $c$ and $d$ ?

Key Concepts

Partial Order

A partial order is a binary relation over a set that is reflexive, antisymmetric, and transitive. It allows comparing some elements within the set, but not necessarily all pairs. In the context of positive integers with divisibility, it means that one integer is related to another if it divides the other without a remainder.

Divisibility Relation

Divisibility is a relation where one number divides another evenly if there is no remainder. When expressed as a partial order on positive integers, it defines an ordering where a divides b if b is a multiple of a, establishing a hierarchy of the numbers based on their factors.

Greatest Lower Bound (Infimum)

The greatest lower bound of two elements in a partially ordered set is the largest element that is less than or equal to both of them. It represents the most specific common property shared by both elements. In this divisibility order, this concept is realized by the greatest common divisor.

Least Lower Bound (Greatest Common Divisor - GCD)

The greatest common divisor (GCD) of two positive integers is the largest integer that divides both numbers without leaving a remainder. In the divisibility partial order, it serves as the greatest lower bound because it is the largest number that is a divisor of both integers.

Least Upper Bound (Supremum)

The least upper bound of two elements in a partially ordered set is the smallest element that is greater than or equal to both. It represents an element that is as small as possible while still being above both members in the ordering. In the context of divisibility, this notion is embodied by the least common multiple.

Least Upper Bound (Least Common Multiple - LCM)

The least common multiple (LCM) of two positive integers is the smallest integer into which both numbers divide evenly. In the ordered structure based on divisibility, it acts as the least upper bound because it is the smallest number that can be divided by both integers without leaving a remainder.

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