Question

If \(p\) and \(q\) are positive integers, then \(\\{p n: n \in \mathbb{N}\\} \cap\\{q n: n \in \mathbb{N}\\} \neq \varnothing\).

Short answer

The multiples of two positive integers \(p\) and \(q\) always have at least one number, \(p*q\), in common. Therefore, \(\{p n: n \in \mathbb{N}\} \cap \{q n: n \in \mathbb{N}\} \neq \varnothing\).

Step-by-step solution

Understand the task

The goal is to demonstrate that there is an integer which is a multiple of both \(p\) and \(q\). So, let's consider \(p*q\).

Recognize \(p*q\) as an element of both sets

\(p*q\) is an element of both sets \(\{p n: n \in \mathbb{N}\}\) and \(\{q n: n \in \mathbb{N}\}\) because \(p*q = p*(q*1)\) and \(p*q = q*(p*1)\) where \(q*1\) and \(p*1\) are positive integers, hence \(n\in \mathbb{N}\).

Conclude

Since \(p*q\) is a member of both the set \(\{p n: n \in \mathbb{N}\}\) and \(\{q n: n \in \mathbb{N}\}\), these two sets are not disjoint and their intersection is not empty.

Key concepts

Positive Integers

Positive integers, also known as natural numbers or counting numbers, are the numbers that we use for counting and ordering in everyday life. They begin with 1 and go on indefinitely: 1, 2, 3, and so on. Unlike negative numbers, fractions, or decimals, positive integers are whole numbers greater than zero.

In the context of set and number theory, positive integers play a crucial role because they are the building blocks of more complex numbers and are used to define various mathematical structures and properties. In the given exercise, we consider the multiples of two positive integers, which are themselves positive integers. The concept of multiples is fundamental in arithmetic and number theory, and it is defined as a product of any positive integer and another integer.

Set Theory

Set theory is a fundamental branch of mathematics that deals with the collection of objects, known as sets, and the relationships between these objects. In simple terms, a set is a group of distinct elements that can be anything: numbers, people, letters, etc.

One of the core operations in set theory is the intersection, denoted by the symbol \(\cap\). The intersection of two sets is a new set that contains all the elements that are common to both original sets. When sets have no elements in common, they are considered disjoint, and their intersection is an empty set, denoted by \(\varnothing\).

In the exercise, we're examining the intersection of sets of multiples of two positive integers. The solution shows that the intersection of these two sets of multiples is not empty, indicating that there is at least one number that is a multiple of both positive integers.

Number Theory

Number theory is a branch of mathematics that studies the properties of integers and related concepts. It deals with patterns, relationships, and theorems that emerge from the study of whole numbers. Topics such as divisibility, prime numbers, and greatest common divisors are central to number theory.

In the scenario given in the exercise, an application of number theory is present in the understanding that any two positive integers have a common multiple – a number that is a multiple of both integers. This is clearly demonstrated with the example of the product of the two positive integers, which evidently belongs to the set of multiples for each individual integer. This aspect of number theory supports the idea that the intersection of multiples of any two positive integers is never empty since the product of the two numbers is always included.