Definition of Sets, set braces, and notations.
What is Discrete math?
Discrete mathematics is the study of mathematical structures that are countable or otherwise distinct and separable. Examples of structures that are discrete are combinations, graphs, and logical statements. Discrete structures can be finite or infinite.
Discrete mathematics is in contrast to continuous mathematics, which deals with structures that can range in value over the real numbers, or have some non-separable quality, refer to this link.
Since the time of Isaac Newton and until quite recently, almost the entire emphasis of applied mathematics has been on continuously varying processes, modeled by the mathematical continuum and using methods derived from the 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 more and more important.
The video I used for illustration.
The topic that we will discuss by god’s will is the definition of set. and the other relevant items that come out of the set. What are the most important items?
The first item is the natural numbers. The second item is the whole number.
The third item is the integers. The fourth item is rational and irrational. The fifth item is the real numbers. The next step is how to round numbers. Decimal rounding, or decimal point. Rounding to whole numbers.
Rounding to scientific figures. I hope that subject will find your satisfaction. That was a brief discussion of the content of the video. The video has a subtitle and a closed caption in English.
The list of items included.
The five items are to be included in our discussion. The first item is the natural numbers. The second item is the whole number. The third item is the integers. The fourth item is rational and irrational. The fifth item is the real numbers. The next step is how to round numbers. Decimal rounding, or decimal point. Rounding to whole numbers. Rounding to scientific figures.
Definition of sets.
What is a set? The set is a well-defined collection of objects or ideas. The sets are well-defined, a collection of objects or ideas. These elements for which a certain condition is selected and well defined.
For instance, we could specify that our set has x bigger than or smaller than a certain value, or even x= square of a number value. What is well-defined? A set is well-defined if there is no ambiguity. As to whether or not an object belongs to it, this is well defined to guarantee a correct choice for proper elements. If the element follows the conditions that are stated or not for proper selection between alternatives, exclude the non-appropriate choice.
What are the set braces?
Set braces, this symbol is shown as two curly brackets The symbols which are used to indicate sets.
For example the item you wear shoes, socks, a hat 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. An example is a set of items that someone wears.
The next item is the notation I quote, there is a fairly simple notation for sets. We simply list each element or member separated by a comma. First, we put the curly brackets.
Then, mention the well-defined element, and list all the required elements separated by a comma, that satisfies your statement. and if the number of elements is great, we can use use the three dots which means it goes forever. the same as infinity. Elements are separated by commas.
Three dots means go forever. For this example, the order is not important.
What are the whole numbers?
We start with the definition of terms we have talked about in the introduction. What are the whole numbers? The whole numbers are simply the numbers 0,1,2,3,4,5 and three dots.
The three dots are extended numbers, or so on. As we can see in the graph in a form of a roller, there is no fraction or negative values. An example was given for a set 0,7, 212, and 1023.
The first number is zero, so it is a whole number, also 7 and 212 are positive numbers also considered whole numbers. But for 1/2, it is a fraction, so not a whole number.
The Whole numbers don’t include negative numbers, fractions, or decimals, so 1.1 is a decimal, not a whole number and 3.5 is a decimal so not a whole number. The next item is the Natural number.
The Natural number is the counting number, we start to count, and you start with 1,2,3,4, 5,6, and so on. So we can include that the natural number or the counting number is the whole number after excluding the zero. also does not include a decimal.
can say 1,2,3,4,5,6 Also does not include fractions or negative signs. the starting number is 1, while for the whole number, the starting number is zero., while the starting number in the whole number is 0. So he wrote 1,2,3,4,5 and dots, which means go to infinity.
Also can be reduced to less than -4 and keep decreasing.
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, such that x is less than 10, which means all the numbers to be less than 10, it is written as the set of all x, such that x is an element of the natural number and x is less than 10. 1,2,3,4,5,6,7,8,9.
We cannot increase further, so 9 is the last number. The last number is 9 less than 10. The condition is for numbers less than 10 The natural numbers do not include 0.
We start with number 1 and stopped at number 9 10 is not included, because the next number will be 10, and the condition is for numbers That is why the definition of a set is written as well defined.
How to express the set by symbols? or the technical term, there are symbols assigned to the previous items that we have discussed. Z plus is the symbol given for positive integers and the vertical line means such that all the positive numbers. Integers, x it can be read as x such that integers we will discuss shortly, same as 1,2,3,4, and so on.
The symbol Z is written = then braces, followed by x, as Z= x such as all the number of x such that x is an integer.
The Z is the symbol given for integers, Which includes all the positive numbers and the negative numbers, and also zero. extended from the right and left, it can increase more than, can be 5, 6 to infinity.
symbol Q includes the rational number, he 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 real numbers. While the Φ is a symbol for all empty sets. the set braces are located and inside nothing is there.
For a useful external site, math is fun, introduction to sets math is fun.
For the next post, Definition of Natural numbers, Integers.