The Language of Sets (Part 1) | Math in the Modern World | Aikz Adonis

Hello class! Welcome back to my channel. Again, I am Ives Adonis, your teacher in Mathematics and the Modern World. In today's learning episode, we will have a continuation of the topic Mathematical Language and Symbols. But this time, we will focus on the language of sets. Are you now ready? So, let's start this episode with a game. Are you familiar of four picks one word? I know you do. In this game, you are going to see four pictures and you will give me one word that will best describe the pictures. So, can we now start? Let's start! Okay, time's up. So, the correct word is set. If you got it right, congratulations! Now, let's see where the correct word is set. The first picture shows a set of basketball players. They are the famous characters in Slam Dunk. And my favorites are Hanamichi Sakuragi and Urukawa. The second picture is a salad set, while the third picture are the kitchenware set. And finally, the fourth picture is a set of clothes. If you can see, this or set are collection of objects or persons that are related to each other. That's why the word that best describe the four pictures is set. Similarly, in mathematics, a set is also a collection of objects. But a well-defined collection of objects. When we say well-defined, we can easily give the objects or elements of a set. To understand it more, let's have a counterexample of well-defined. Let's say a set of cute students. This set is not well-defined because we have a different perspective of cute. Someone may be cute for me, but not cute for you. Or not cute for me, but looks cute for you. A set that is well-defined has no different perspectives. A set is usually presented by capital letters. For example, Set A and Set B. And the objects that belong in a set are the elements or the members of the set. For example, in Set A, the elements are A, B, C, D, and E. While in Set B, the elements are 1, 2, 3, 4, and 5. Sets are also denoted by braces and commas. The symbol for braces are in color pink in the example, while the commas are in blue-green. Now, let's have the notation of sets. This notation in color blue means x is an element of S. While the second notation in color yellow means that x is not an element of S. You can easily see if the notation is not an element because of the slash or the dash line. For example again, in Set B, 1 is an element of B because 1 is in B. While 6 is not an element of B because 6 is not in B. Another, 2 is an element of B, but 7 is not an element of B. In other words, the elements of B are only the numbers 1 to 5, and the other numbers are not the element of B. In example number 2, C is the set of all positive even numbers from 1 to 20. Remember that even numbers are numbers that are divisible by 2. So, the elements of C will start from 2 and ends in 20. All numbers that are written here are elements of C, while the numbers that are not written here, though they are part of 1 to 20, are not elements of C. And this, or those even numbers that are not in the interval of 1 to 20, are also not an element of C. A variation of notation is used to describe a very large set. For example, I have here 1, 2, 3, and so on up to 100. So, it refers to a set of all integers from 1 to 100. As you can see, not all integers from 1 to 100 are written. Instead, we use an ellipsis to represent the numbers 4 to 90. Another example, the set of all positive numbers. This is somewhat the same as in the first example, but we remove the 100 at the end because all positive numbers has no end. The ellipsis here denotes that this set of numbers will approach positive infinity. The third example is the set of all negative integers. As you can see, the ellipsis is on the left and the element starts with negative 1. It denotes that the set of numbers will approach negative infinity. Again, this symbol, three dots, is called an ellipsis and is read and so forth. Ellipsis is used if the element in a set is so large. There are several ways to describe a set. These are listing method or roaster method, set builder notation, and interval notation. Now, let's discuss first the listing method or the roaster method. And this method will list all the elements in a set between braces, if possible. For example, the set of all vowels in an English alphabet is represented by the set of S wherein the elements are A, E, I, O, U. As you can see, elements which are the vowels of an alphabet are listed between braces. Another example is the set of all add numbers less than 10. Remember, add numbers are whole numbers that 1 divided by 2 leaves a remainder of 1. Since we are referring to add numbers less than 10, the elements that we can list between are 1, 3, 5, 7, and 9. Sometimes, we do not list all the elements of a set. Instead, we list only some of the elements and then use ellipsis to describe a very large set, just what we discussed in the previous slide. Now, let's go back again to the previous examples. In this first example, we do not list all the integers from 1 to 100. Instead, we just list the first three elements 1, 2, 3, and use ellipsis to represent the integers from 4 to 99, and then list the last integer in a set, which is 100 in this case. While in the second example, we do not list the last element because the set of all positive integers has no definite end. It's actually approaching positive infinity. And in the third example, we start from the right going to the left because we are approaching negative infinity. Remember that the highest negative number is negative 1 followed by negative 2 and decrease as it moves going to the left. In this last example, you can see that ellipsis is used on both sides of a set. This is because ellipsis can be used in different order depending on the pattern of set. The ellipsis in this example pertains that the elements of this set will approach both negative infinity and positive infinity. So, is listing method or roaster method clear to you? Good. Now, let's move to the second way of describing a set, which is set builder notation. Set builder notation. In this way, we characterize all the elements in a set by stating the property or properties they must have b member. For example, all x such that x is an even number less than or equal to 10. This line is read such that. So, what are the possible values of x in this set builder notation? Yes, the values of x are 2, 4, 6, 8 and 10. As you can see, this way of writing a set is the listing or the roaster method. This only shows that we can use any of these ways to describe and represent a set. Another example is the set Q containing all x such that x is a rational number. Take note that rational numbers are numbers that can be written in the form A over B such that negative three-fourth, negative one-half, one-half, three-fourth, five-sixth and many more. Since it is hard to list all the rational numbers, we prefer to use the set builder notation to describe the set Q. Another, let's have more examples of sets in set builder notation. So, for the first example, this set is read as the set of all elements x in the set of integers such that x is greater than negative three but less than positive six. So, what are the elements in this set? Take note that x is greater than negative three but less than positive six. So, the elements are between negative three and six and these are negative two, negative one, zero, one, two, three, four and five. Now, if we will wrap this set in a number line, you will see that negative three and six are not included in the elements. So, we put there open circles. Open circle denotes that the element is not included in the set. This open circle is used if the inequality symbol is either greater than or less than. Now, let's have another example. This set is read as the set of all elements x in the set of integers such that x is greater than or equal to negative three but less than or equal to positive six. So, what are the elements again of this set? As you can see, the inequality symbols used are greater than or equal to and less than or equal to. So, you can easily say that negative three and six are part of the elements in a set because of the inequality symbols greater than or equal to and less than or equal to. And if we will wrap this set in a number line, you will see that negative three and positive six are included in the elements of the set. So, we use closed circle or solid circle. Closed circle or solid circle is used when the inequality symbol is either greater than or equal to or less than or equal to. And aside from writing a set in a set builder notation, we can also write a set in interval notation. Recall that there are nine notations for the interval of real numbers and these are the interval notations. So, this in circle notations. It is also shown in the table the names of the interval, their inequality or description of set builder notation, and of course their graphs. If you will observe the relationship among the notation, inequality symbols used, and kinds of circle in the graph, you will see that open or closed bracket is used when the inequality symbol used is greater than or equal to or less than or equal to. And you already learned a while ago that if the inequality symbol used is greater than or equal to or less than or equal to, we use closed circle or solid circle. On the other hand, we use open or closed parenthesis when the inequality symbol used is greater than or less than. And if these are the inequality symbols, we use open circle or hollow circle. Okay class, so please put in your mind that brackets means included in the set while parenthesis means not included in the set. But if you will observe the notation with negative infinity and positive infinity, you will see that it always use parenthesis. The inequality symbols also are greater than or less than. And to really understand the interval notation of a set, let's have these examples. And this interval notation, open bracket, three, comma, seven, and then closed bracket, we know that three and seven are included in a set because again, bracket means included. So if we will list all the elements in the interval notation, or in this interval notation, we will put the elements three, four, five, six, and seven inside the braces. Again, these are the braces while these are the brackets. Now, what if the symbol beside three is an open parenthesis? Do you think three is included in the set? No, it is not included. How about seven? Yes, it is included because the symbol beside seven is a bracket. Therefore, the elements of this interval notation are four, five, six, and seven. How about this one? Beside three is a bracket while beside seven is a parenthesis, which is included and which is not? Yeah, you're right. Three is included in the set while seven is not included. And last one, what if the symbols used are both parentheses? Of course, both three and seven are not included in the set. So, is it clear to you when to use the parentheses and the brackets? Very good. Now, let's talk about the cardinality of a set. Cardinality refers to the number of elements in a set. We can denote cardinality of set A by either of these two symbols. For example, set A with elements negative one, zero, one, two, three, four, and five. So, what is the cardinality of this set A? To get its cardinality, we will simply count the number of elements in set A. One, two, three, four, five, six, seven. So, the cardinality of set A is seven. How about this second example? Set B containing all vowels in an English alphabet. To find the cardinality, we need to count the number of vowels in an English alphabet. The vowels in an English alphabet are A, E, I, O, U. Therefore, the cardinality of set B is five. Speaking of cardinality, we can only count the elements of finite sets. Finite sets are sets which either has no element or has elements which could all be possibly listed or counted. The first two elements or the first two examples a while ago are finite sets because we are able to count the number of its elements or its cardinality. How about this one? Set C containing all natural numbers less than 10. Is this a finite set? Yes, it is. Natural number are numbers from one to positive infinity. Since we are just talking to natural numbers less than 10, it is countable. So, what is the cardinality of set C? Very good! The cardinality of set C is nine because the elements are 1, 2, 3, 4, 5, 6, 7, and 9 only. And take note that 10 is not included because set of all natural numbers less than 10. How about infinite sets or infinite sets? Infinite sets are sets whose elements could not all be possibly listed or counted. For example, the set of whole numbers, the set of integers, and the set of real numbers. So, these are examples of infinite sets. The elements of the first set W approaches positive infinity while the set of integers and real numbers are both approaching or approaches positive infinity and negative infinity. We also have null set. A null set is a set with no elements. Null set can be denoted by either of these symbols. It is also called empty set. For example, set A and B are obviously null sets while set C is also a null set because there's no integer between 3 and 4. When we say integer, it must be whole number and can be written without fractional component. So, 2.1 is not an integer. 2.5 is also not an integer. Therefore, there's no integer between 3 and 4. That's why set C is a null set. Now, let's talk about equality of sets. Let A and B be sets. If both A and B have the same elements, then set A is equal to set B. For example, set A is equal to set B because all the elements in A is also in B. You might think of its order but in set, the order of the elements doesn't matter. What matters is the elements in the set. How about in this example? You may think that set A has more elements in set B but notice that the elements in A are just the same as in B. Take note that in set, the elements that repeat can be written as a single element. So here, the two 1s can be written only once and the three 2s can be written also once. So, set A is just equal to set B. So, it's the end of the topic this time. So, do you learn from this video? I hope so. By the way, this is just the introduction to sets. 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