Question

Write each of the following sets by listing their elements between braces. $$ \\{x \in \mathbb{R}: \sin \pi x=0\\} $$

Short answer

The members of the set are all integers. Hence, the set can be written as \{x: x is an integer\}.

Step-by-step solution

Understand the set

It is important to understand that the set in question includes all real numbers \(x\) such that \(\sin(\pi x) = 0\). This means we need to find all values of \(x\) which make \(\sin(\pi x) = 0\) true.

Apply properties of Sine function

Given the properties of the sine function, it is known that \(\sin(k\pi) = 0\) for all integers \(k\). Therefore, we must find all values of \(x\) such that \(\pi x\) is an integer multiple of \(\pi\).

List the members of the set

From step 2, it's clear that the solution to the problem \(\pi x = k\pi\) (where \(k\) is any integer) will be \(x = k \) for all integers \(k\). Hence, the set can be represented as \{x: x is an integer\}.

Key concepts

Properties of the Sine Function

The sine function, which is frequently used in trigonometry, has several key properties that are crucial for solving various mathematical problems. One fundamental property is its periodicity; the sine function repeats its values in regular intervals known as periods. For the sine function, the period is \(2\pi\), which means that \(\sin(\theta) = \sin(\theta + 2\pi k)\) for any integer \(k\).

Another important property is that the sine function is zero at multiples of \(\pi\). Specifically, \(\sin(k\pi) = 0\) for every integer \(k\). This property is essential when solving equations like \(\sin(\pi x) = 0\) because it allows us to conclude that \(\pi x\) must be an integer multiple of \(\pi\) - this narrows down the values of \(x\) to just integers. The function is also symmetric about the origin, which implies that \(\sin(-\theta) = -\sin(\theta)\), reflecting its odd-function characteristics.

Real Numbers

The set of real numbers, denoted by \(\mathbb{R}\), includes all the numbers on the continuous number line. It consists of rational numbers (like \(\frac{1}{2}\), \(3\), and \(-4\)) as well as irrational numbers (like \(\sqrt{2}\) and \(\pi\)). One fundamental feature of real numbers is that they can be as close to each other as possible, without any 'gaps' between them. This concept is known as density.

The understanding of real numbers is vital in many areas of mathematics, including calculus, algebra, and set theory. In set notation problems involving real numbers, it's important to recognize which subsets of real numbers are being described, like the set of all integers or the set of all positive real numbers.

Set Notation

Set notation is a system of symbols and visual representations used in mathematics to precisely define collections of objects. The most basic form of set notation is listing the elements between braces: \(\{a, b, c\}\). Another common way to describe a set is by using a set-builder notation which involves a condition that members must satisfy. For instance, the set \(\{ x \in \mathbb{R} : P(x) \}\) includes all real numbers \(x\) for which the property \(P(x)\) is true.

Sets can be finite, with a countable number of elements, or infinite if the elements cannot be counted. An understanding of set notation is fundamental for expressing mathematical ideas clearly and concisely, and for understanding the relationships between different sets, such as subsets, intersections, and unions.