Last Updated on September 10, 2026 by Maged kamel
Definition of Sets, set braces, and notations.
What is Discrete math?
Discrete mathematics studies mathematical structures that are countable or otherwise distinct and separable. Examples of discrete structures include combinations, graphs, and logical statements. Discrete structures can be finite or infinite.
Continuous mathematics, in contrast to discrete mathematics, deals with structures that can take values in the real numbers or exhibit non-separable properties. Refer to this link.
From Isaac Newton until recently, applied mathematics has largely emphasized continuously varying processes, modeled by the mathematical continuum and using methods derived from differential and integral calculus. In contrast,
discrete mathematics concerns itself mainly with finite collections of discrete objects. With the growth of digital devices, especially computers, discrete mathematics has become increasingly important.
The list of items included.
We include five items in our discussion. The first item is the natural numbers. The second item is the whole numbers. The third item is the integers. The fourth item is rational and irrational numbers.
The fifth item is the real numbers. The next step covers how to round numbers, including decimal rounding, rounding to whole numbers, and rounding in scientific notation.
Definition of sets.
What is a set? A set is a well-defined collection of objects or ideas. Sets are well-defined collections of objects or ideas. Elements are objects for which a specific condition is selected and well-defined.
For instance, we could specify that our set contains x greater than or less than a specific value, or even x equal to the square of a number.
What is well-defined? A set is well-defined if there is no ambiguity. Whether an object belongs to it is well-defined, guaranteeing a correct choice of elements.
If the element follows the stated conditions, select it; otherwise, exclude the inappropriate choice.
A set is a well-defined collection of objects or ideas. Elements are selected based on specific conditions and well-defined criteria. For example, a set could specify objects that are greater than, less than, or equal to a particular value, or have an exact mathematical relationship.
A set is well-defined when there is no ambiguity about whether an object belongs. This ensures you select the correct elements by excluding non-conforming choices.
What are set braces?
Set braces: These symbols are shown as two curly brackets. They indicate sets.
For example, you wear shoes, socks, a hat, a shirt, pants, and so on. I quote, “I’m sure you could come up with at least a hundred.” This is known as a set. For example, a set is a collection of items someone wears.
The following item is the notation I quote; there is a reasonably simple notation for sets. We list each element, separated by commas. First, we put the curly brackets.
Then, mention the well-defined element and list all the required elements separated by commas that satisfy your statement. If the number of elements is infinite, we can use the three dots, which means it goes on forever, just like infinity—Commas separate elements.
Three dots mean it goes on forever. For this example, the order is not essential.
What are the whole numbers?
We begin with the definitions of terms that we discussed in the introduction. Whole numbers are simply 0, 1, 2, 3, 4, 5, and so on. The three dots signify that the numbers continue indefinitely.
In the graph, there are no fractions or negative values. For example, a set could be {0, 7, 212, 1023}. Zero is a whole number, and so are 7 and 212, which are positive whole numbers. However, 1/2 is a fraction, not a whole number.
Whole numbers do not include negative numbers, fractions, or decimals. For example, 1.1 and 3.5 are decimals and not whole numbers. The following item is a natural number.
Natural numbers are counting numbers that start from 1 and include 2, 3, 4, 5, 6, and so on. Unlike whole numbers, natural numbers do not contain zero or decimals. The next slide shows the differences between integers, rational numbers, and Real numbers.
The next slide shows the difference between rational and irrational numbers.
Set-builder illustration, quoted from Math is Fun.
We will proceed to a new slide. This is the set-builder: how to pronounce the Expression and write it. For instance, write the left brace, then include x, then the vertical line followed by x, which is an element that belongs to, followed by the symbol N for natural numbers.
For example, x is less than 10, which means all the numbers are less than 10; we write this as the set of all x such that x is a natural number and x < 10. 1,2,3,4,5,6,7,8,9.
We cannot go any higher, so 9 is the last number. The last number is 9, which is less than 10. The condition is for numbers less than 10. Natural numbers do not include 0.
We start at 1 and stop at 9. 10 is not included because the next number is 10, and the condition is for numbers less than 10. That is why the definition of a set is written as well-defined.
How do we represent the set symbolically? In other words, we assign symbols to the items we discussed earlier.
Z+ is the symbol given for positive integers, and the vertical line means that all the positive numbers are included. Integers, x, can be read as x, such that x is an integer, just like 1, 2, 3, 4, and so on.
The symbol Z is written as Z = {x: x is an integer}, i.e., all numbers x such that x is an integer.
Z is the symbol used for the set of integers, which includes all positive and negative numbers and zero. It extends to the left and right, including numbers greater than 5, 6, or even infinity.
Symbol Q includes rational numbers; it introduces the braces and x such that x is a Rational Number.
While the symbol R is for real numbers between braces, x is all numbers such that each element belongs to the real numbers. In contrast, the symbol Φ symbolizes the empty set.
The set braces are located, and inside, there is nothing.
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For a useful external site, try Math is Fun and an introduction to sets.
For the next post, the definition of Natural Numbers and Integers.