PRECALCULUS: Reminder of Sets

This page is a listing of the main topics related to sets.

Definition of a Set A set is a collection of objects or things or even concepts, called elements. In a real sense, a set is a group of things where we have some way to determine if something is either in the group or not in the group. We might be able to write out the names of all the elements of the set. If we can do that, then someone else can look at our list of elements and can determine if something is or is not in our set. For example, I might tell you that I am thinking of a set whose members are the colors red, blue, and yellow. My set has three elements. You can read the list of those elements and you immediately know that "blue" is in the set and that the color "purple" is not in the set.
Alternatively, we may describe a set by giving a rule for determining iof something is a member of the set. For example, I might tell you that I am thinking of the set of all counting number whose English names are exactly three letters long. It is not immediately clear just how many elements are in this set, but we can use the rule to test out different possible values. The number "10" is in the set because "ten" satisfies the rule, it has exactly three letters in it. On the other hand, the number "8" is not in the set because "eight" has five, not three, letters in it. If we think about the counting numbers we can eventually change the rule into a list of the numbers "1", "2", "6", and "10".
An interesting consequence of our definition of a set is that something is either an element of the set or it is not an element of the set. Furthermore, if something is an element, then it is only listed one time in the set. That is, we do not repeat elements in a set.

Listing a set Gives the list of elements of the set, separated by commas, enclosed in braces, as in {4,6,8,10} as a set. Notice that there is no order to the elements of a set. The set given as {4,6,8,10} is the same as the set given as {6,10,4,8} which is the same as the set given as {8,4,6,10}.
However, even though the elements of the set can appear in any order, there are times that we find it convenient to list the elements in a specific order. If a set is large and if its elements form some pattern, then we can use three dots (called an ellipsis) to indicate that the pattern continues either to a terminating few elements, as in {3,6,9,12,...,57,60}, or forever, as in {1,4,9,16,25,36,...}.
The ellipsis can be used at either side, or on both sides of the pattern. Threfore, we understand
{1, 3, 5, 7, 9, ...} to continue with 11, 13, 15, 17, and so on. And we understand {..., 5, 7, 9, 11} to contain the numbers 3, 1, -1, -3, and so on, in addition to the numbers listed, 5, 7, 9, and 11. And, we understand { ..., 1/8, 1/4, 1/2, 1, 2, 4, 8, ... } to contain values such as 1/16, 1/32, 1/64, and so on, as well as 16, 32, 64, 128, and so on. As noted above, the ellipsis can be used in the middle of a listing in order to shorten the list. An example of this would be {31, 32, 33, 34, ..., 91, 92, 93} where the ellipsis saves us from having to write out the numbers from 35 through 90.

Set-builder notation. Specifies the elements of a set by giving a rule that is used to determine if something is an element of the set, as in { x | x is a registered student in a WCC math class on April 7, 1998} where the vertical bar, "|", is read as "such that". Thus the set described above is read as The set of all things "x" such that "x" is a registered student in a WCC math class on April 7, 1998.
The set-builder notation has both advantages and disadvanages over the "listing" method for defining a set. With the set-buildeer notation we need only state a rule that allows us to test to see if something is or is not an element of the set. The example given above,
{ x | x is a registered student in a WCC math class on April 7, 1998} is just such a rule. The advantage is that we do not hve to list all of the elements of the set. For this example, that list would include over 2000 people. Rather than the long list, we have a short rule. Furthermore, by stating the rule we are conveying the "organizing concept" behind the set. If we had written the list of over 2000 names it would not be clear, from the list, why someone is on the list and why someone else is not. The fact that the set-builder reveals the "organizing concept" behind the set makes this form appealing.
The disadvantage of set-builder notation is that it relies on our language and on our understanding of that language to apply the rule for set membership. For example, there is some room for disagreement about the meaning of even such a simple statement as
is a registered student in a WCC math class on April 7, 1998 What if a student drops a math class at 3:30 on April 7? Is that student registered or not on April 7? What is a student enrolls on April 7? What if a student is registered for a non-credit Math class? What do we do with students who are registered for a math class but where that class only met during the first half of the term, and therefore, it was over before March? Thus, even though the intention of the set-builder notation may have been clear, it is possible to raise questions about the rule. This is different from the set listing method. There, for the most part, it is absolutely clear what is and is not a member of the set.

Naming sets We often use capital letters to name a set, as in B={4,5,6,7}. One problem with this is that we quickly run out of capital letters. Therefore, we end up reusing letters from problem to problem. Whenever we make reference to a set B you need to look back within the problem or the text to find out just what set named B we are using.

Set equality Two sets are equal if they have exactly the same elements. As you might expect, we use the equals symbol, "=", to denote set equality. Therefore, the statement A=B is read as "the set A is equal to the set B", meaning that the two sets have exactly the same elements.

Subsets Set A is a subset of set B if every element of A is also an element of B. We use the symbol to indicate "is a subset". Thus, AB is read as "A is a subset of B". Notice that if AB then it is possible that A=B. If A=B then certainly every element of A must also be an element of B. But that is the definition of being a subset. Therefore, A is a subset of B. Of course, if A=B, then every element of B is also an element of A. Therefore, it is also true that BA.
The symbol for "is a subset of" uses the lower line to remind us that the two sets could be equal.
We might notice that if AB and BA then it must be true that A=B. In many cases, when we want to prove that two sets are equal, we instead prove that the first set is a subset of the second, and that the second is a subset of the first.
We use the related symbol to mean "is not a subset of". If we say that DE we are saying that D is not a subset of E. In particular, this means that there must be at least one element of D that is not in E.

Proper Subsets Set A is a proper subset of set B if A is a subset of B and if B has at least one element that is not in A. We use the symbol to indicate "is a proper subset". Thus, AB is read as "A is a proper subset of B".
We use the related symbol to mean "is not a proper subset of". If we say that DE we are saying that D is not a proper subset of E. There are two ways for set D to not be a proper subset of set E. First, there may be at least one element of D that is not in E. Second, D would not be a proper subset of E if D=E. Either approach could be true.

Empty Set or Null Set There is a special set called the Empty set or the Null set. It is the set with no elements. We use two different symbols for the empty set. One is braces with nothing between them, as in { }. The other symbol used to represent the empty set is . Note that the empty set is a subset of every set because every element in the empty set (and there are none) is also an element of the other set. Thus, for any set A we can write A. Either symbol, { } or , can be used to represent the empty or null set. However, note that the set {} is not the empty set. Rather {} is a set with one element, and that element is the empty set.

Universal set One might expect the universal set to be the set of everything, the set with every possible element in it, just as the empty set is the set with no elements. Although we could do this, it is better for us to define the Universal set as the set of all things under current consideration. This means that the Universal set changes from problem to problem, but it also means that the Universal set is reasonably small and well defined for each problem. We use the capital letter U to represent the Universal set.

Complement of a set. If the Universal set is the set of all the elements under consideration, and A is a set of some of those elements, then the set of all elements in the Universal set that are not in A is called the complement of A, and is denoted as A'. This is an important concept. It means that for any universal set, any subset, A, of the universal set really divides the universal set into two subsets, A and A'. Furthermore, an element is either in A or it is in A', it can not be in both of them at the same time.

Is an element of We use the symbol to represent "is an element of". Thus, for the set A={dog,bird,cat,horse,cow, mouse} we could write cow A to say "cow is an element of A.

Is not an element of We use the symbol to represent "is not an element of". Thus, for the set A={dog,bird,cat,horse,cow, mouse} we could write rat A to say "rat is not an element of A.

Intersection The intersection of two sets is a new set that has all of the elements that are common to the original two sets. That is, to be in the Intersection, an element must be in both of the original sets. We use the symbol to mean "intersect" and we write AB to say A intersect B. If A={2,4,6,8,10,12,14,16} and B={3,6,9,12,15,18,21,24,27} then AB={6,12}.

Disjoint Sets Two sets are said to be disjoint if the intersection of the two sets is the empty set. In symbols, if AB= the A and B are called disjoint. It should be clear that A and A' (the complement of A) are always disjoint.

Mutually disjoint For more than two sets, if each pair of sets is disjoint, then the sets are said to be mutually disjoint.

Union The union of two sets is a new set that has all of the elements that are elements of either of the original two sets. That is, to be in the union, an element must be in one or the other, or both, of the original sets. We use the symbol to mean "union" and we write AB to say A union B. If A={2,4,6,8,10,12,14,16} and B={3,6,9,12,15,18,21,24,27} then AB={2,3,4,6,8,9,10,12,14,15,16,18,21,24,27}. Also, note that for AA'=U.

Venn Diagrams A Venn diagram is a picture that shows the relationship between sets. In general we use a rectangle to represent the Universal set for the problem, and we show various sets (subsets of the Universal set) as cirlces within the rectangle. If two circles overlap, then the two sets have a non-empty intersection. If two circles do not overlap, then the two sets are disjoint. If one circle is entirely inside another, then the first set is a subset (in fact a proper subset) of the second.

Cardinality of a set The cardinality of a set is the number of elements in the set. We use the notation n(A) to represent the "cardinality of set A". If A={2,4,6,8,10,12,14,16} and B={3,6,9,12,15,18,21,24,27} then n(A)=8 and n(B)=9.

Finite sets To most of us it is pretty clear that the set {3, 4, 5, 6} is finite. After all, it has just 4 elements. The set {2, 4, 6, 8, 10, ..., 98, 100} is finite since it has exactly 50 elements. The set {2, 4, 6, 8, 10, ..., 996, 998, 1000} is finite; it has exaclty 500 elements. A set is finite if we could theoretically count the elements and end our count.
Consider the following situation. We go to our local food store and we buy one pound box of salt. Now that we have a specific box of salt, we can look at the the set of all grains of salt in the box. Clearly there is a huge number of grains of salt in that box. I would not want to have to count them. But I could. And if I did, I would arrive a some specific number of grains of salt in the box. Therefore, the set of all grains of salt in that box is a finite set.
If I bought a case of boxes of salt, then the set of grains of salt that I bought would be extremely large, but still finite. In fact, the set of grains of salt in all the boxes of salt in all the kitchens in all the houses in all of Washtenaw County at this moment, is a finite set. Now, there is no way that we can stop time and gather up all the boxes of salt in all the kitches in all the houses in all of Washtenaw County so that we can actually count the grains of salt, but theoretically we could do this, and if we did we could count the grains and at some point we would have counted each grain and we could stop. Therefore, our set is finite.

Infinite sets A set that has an unlimited number of elements is called an infinite set. For example, the set of all even counting numbers, {2, 4, 6, 8, 10, ...} is infinite. It does not stop. The set of all numbers that use the character 7 is infinite. It has elements such as 7, 17, 27, 70, 71, 234789, and 7827927 in it, along with an endless collection of other numbers.
Strangely enough, there are very few infinite sets in the real world. Most of the inifinte sets that we will deal with are sets of numbers.

Definition of a Set	A set is a collection of objects or things or even concepts, called elements. In a real sense, a set is a group of things where we have some way to determine if something is either in the group or not in the group. We might be able to write out the names of all the elements of the set. If we can do that, then someone else can look at our list of elements and can determine if something is or is not in our set. For example, I might tell you that I am thinking of a set whose members are the colors red, blue, and yellow. My set has three elements. You can read the list of those elements and you immediately know that "blue" is in the set and that the color "purple" is not in the set. Alternatively, we may describe a set by giving a rule for determining iof something is a member of the set. For example, I might tell you that I am thinking of the set of all counting number whose English names are exactly three letters long. It is not immediately clear just how many elements are in this set, but we can use the rule to test out different possible values. The number "10" is in the set because "ten" satisfies the rule, it has exactly three letters in it. On the other hand, the number "8" is not in the set because "eight" has five, not three, letters in it. If we think about the counting numbers we can eventually change the rule into a list of the numbers "1", "2", "6", and "10". An interesting consequence of our definition of a set is that something is either an element of the set or it is not an element of the set. Furthermore, if something is an element, then it is only listed one time in the set. That is, we do not repeat elements in a set.
Listing a set	Gives the list of elements of the set, separated by commas, enclosed in braces, as in {4,6,8,10} as a set. Notice that there is no order to the elements of a set. The set given as {4,6,8,10} is the same as the set given as {6,10,4,8} which is the same as the set given as {8,4,6,10}. However, even though the elements of the set can appear in any order, there are times that we find it convenient to list the elements in a specific order. If a set is large and if its elements form some pattern, then we can use three dots (called an ellipsis) to indicate that the pattern continues either to a terminating few elements, as in {3,6,9,12,...,57,60}, or forever, as in {1,4,9,16,25,36,...}. The ellipsis can be used at either side, or on both sides of the pattern. Threfore, we understand {1, 3, 5, 7, 9, ...} to continue with 11, 13, 15, 17, and so on. And we understand {..., 5, 7, 9, 11} to contain the numbers 3, 1, -1, -3, and so on, in addition to the numbers listed, 5, 7, 9, and 11. And, we understand { ..., 1/8, 1/4, 1/2, 1, 2, 4, 8, ... } to contain values such as 1/16, 1/32, 1/64, and so on, as well as 16, 32, 64, 128, and so on. As noted above, the ellipsis can be used in the middle of a listing in order to shorten the list. An example of this would be {31, 32, 33, 34, ..., 91, 92, 93} where the ellipsis saves us from having to write out the numbers from 35 through 90.
Set-builder notation.	Specifies the elements of a set by giving a rule that is used to determine if something is an element of the set, as in { x \| x is a registered student in a WCC math class on April 7, 1998} where the vertical bar, "\|", is read as "such that". Thus the set described above is read as The set of all things "x" such that "x" is a registered student in a WCC math class on April 7, 1998. The set-builder notation has both advantages and disadvanages over the "listing" method for defining a set. With the set-buildeer notation we need only state a rule that allows us to test to see if something is or is not an element of the set. The example given above, { x \| x is a registered student in a WCC math class on April 7, 1998} is just such a rule. The advantage is that we do not hve to list all of the elements of the set. For this example, that list would include over 2000 people. Rather than the long list, we have a short rule. Furthermore, by stating the rule we are conveying the "organizing concept" behind the set. If we had written the list of over 2000 names it would not be clear, from the list, why someone is on the list and why someone else is not. The fact that the set-builder reveals the "organizing concept" behind the set makes this form appealing. The disadvantage of set-builder notation is that it relies on our language and on our understanding of that language to apply the rule for set membership. For example, there is some room for disagreement about the meaning of even such a simple statement as is a registered student in a WCC math class on April 7, 1998 What if a student drops a math class at 3:30 on April 7? Is that student registered or not on April 7? What is a student enrolls on April 7? What if a student is registered for a non-credit Math class? What do we do with students who are registered for a math class but where that class only met during the first half of the term, and therefore, it was over before March? Thus, even though the intention of the set-builder notation may have been clear, it is possible to raise questions about the rule. This is different from the set listing method. There, for the most part, it is absolutely clear what is and is not a member of the set.
Naming sets	We often use capital letters to name a set, as in B={4,5,6,7}. One problem with this is that we quickly run out of capital letters. Therefore, we end up reusing letters from problem to problem. Whenever we make reference to a set B you need to look back within the problem or the text to find out just what set named B we are using.
Set equality	Two sets are equal if they have exactly the same elements. As you might expect, we use the equals symbol, "=", to denote set equality. Therefore, the statement A=B is read as "the set A is equal to the set B", meaning that the two sets have exactly the same elements.
Subsets	Set A is a subset of set B if every element of A is also an element of B. We use the symbol to indicate "is a subset". Thus, AB is read as "A is a subset of B". Notice that if AB then it is possible that A=B. If A=B then certainly every element of A must also be an element of B. But that is the definition of being a subset. Therefore, A is a subset of B. Of course, if A=B, then every element of B is also an element of A. Therefore, it is also true that BA. The symbol for "is a subset of" uses the lower line to remind us that the two sets could be equal. We might notice that if AB and BA then it must be true that A=B. In many cases, when we want to prove that two sets are equal, we instead prove that the first set is a subset of the second, and that the second is a subset of the first. We use the related symbol to mean "is not a subset of". If we say that DE we are saying that D is not a subset of E. In particular, this means that there must be at least one element of D that is not in E.
Proper Subsets	Set A is a proper subset of set B if A is a subset of B and if B has at least one element that is not in A. We use the symbol to indicate "is a proper subset". Thus, AB is read as "A is a proper subset of B". We use the related symbol to mean "is not a proper subset of". If we say that DE we are saying that D is not a proper subset of E. There are two ways for set D to not be a proper subset of set E. First, there may be at least one element of D that is not in E. Second, D would not be a proper subset of E if D=E. Either approach could be true.
Empty Set or Null Set	There is a special set called the Empty set or the Null set. It is the set with no elements. We use two different symbols for the empty set. One is braces with nothing between them, as in { }. The other symbol used to represent the empty set is . Note that the empty set is a subset of every set because every element in the empty set (and there are none) is also an element of the other set. Thus, for any set A we can write A. Either symbol, { } or , can be used to represent the empty or null set. However, note that the set {} is not the empty set. Rather {} is a set with one element, and that element is the empty set.
Universal set	One might expect the universal set to be the set of everything, the set with every possible element in it, just as the empty set is the set with no elements. Although we could do this, it is better for us to define the Universal set as the set of all things under current consideration. This means that the Universal set changes from problem to problem, but it also means that the Universal set is reasonably small and well defined for each problem. We use the capital letter U to represent the Universal set.
Complement of a set.	If the Universal set is the set of all the elements under consideration, and A is a set of some of those elements, then the set of all elements in the Universal set that are not in A is called the complement of A, and is denoted as A'. This is an important concept. It means that for any universal set, any subset, A, of the universal set really divides the universal set into two subsets, A and A'. Furthermore, an element is either in A or it is in A', it can not be in both of them at the same time.
Is an element of	We use the symbol to represent "is an element of". Thus, for the set A={dog,bird,cat,horse,cow, mouse} we could write cow A to say "cow is an element of A.
Is not an element of	We use the symbol to represent "is not an element of". Thus, for the set A={dog,bird,cat,horse,cow, mouse} we could write rat A to say "rat is not an element of A.
Intersection	The intersection of two sets is a new set that has all of the elements that are common to the original two sets. That is, to be in the Intersection, an element must be in both of the original sets. We use the symbol to mean "intersect" and we write AB to say A intersect B. If A={2,4,6,8,10,12,14,16} and B={3,6,9,12,15,18,21,24,27} then AB={6,12}.
Disjoint Sets	Two sets are said to be disjoint if the intersection of the two sets is the empty set. In symbols, if AB= the A and B are called disjoint. It should be clear that A and A' (the complement of A) are always disjoint.
Mutually disjoint	For more than two sets, if each pair of sets is disjoint, then the sets are said to be mutually disjoint.
Union	The union of two sets is a new set that has all of the elements that are elements of either of the original two sets. That is, to be in the union, an element must be in one or the other, or both, of the original sets. We use the symbol to mean "union" and we write AB to say A union B. If A={2,4,6,8,10,12,14,16} and B={3,6,9,12,15,18,21,24,27} then AB={2,3,4,6,8,9,10,12,14,15,16,18,21,24,27}. Also, note that for AA'=U.
Venn Diagrams	A Venn diagram is a picture that shows the relationship between sets. In general we use a rectangle to represent the Universal set for the problem, and we show various sets (subsets of the Universal set) as cirlces within the rectangle. If two circles overlap, then the two sets have a non-empty intersection. If two circles do not overlap, then the two sets are disjoint. If one circle is entirely inside another, then the first set is a subset (in fact a proper subset) of the second.
Cardinality of a set	The cardinality of a set is the number of elements in the set. We use the notation n(A) to represent the "cardinality of set A". If A={2,4,6,8,10,12,14,16} and B={3,6,9,12,15,18,21,24,27} then n(A)=8 and n(B)=9.
Finite sets	To most of us it is pretty clear that the set {3, 4, 5, 6} is finite. After all, it has just 4 elements. The set {2, 4, 6, 8, 10, ..., 98, 100} is finite since it has exactly 50 elements. The set {2, 4, 6, 8, 10, ..., 996, 998, 1000} is finite; it has exaclty 500 elements. A set is finite if we could theoretically count the elements and end our count. Consider the following situation. We go to our local food store and we buy one pound box of salt. Now that we have a specific box of salt, we can look at the the set of all grains of salt in the box. Clearly there is a huge number of grains of salt in that box. I would not want to have to count them. But I could. And if I did, I would arrive a some specific number of grains of salt in the box. Therefore, the set of all grains of salt in that box is a finite set. If I bought a case of boxes of salt, then the set of grains of salt that I bought would be extremely large, but still finite. In fact, the set of grains of salt in all the boxes of salt in all the kitchens in all the houses in all of Washtenaw County at this moment, is a finite set. Now, there is no way that we can stop time and gather up all the boxes of salt in all the kitches in all the houses in all of Washtenaw County so that we can actually count the grains of salt, but theoretically we could do this, and if we did we could count the grains and at some point we would have counted each grain and we could stop. Therefore, our set is finite.
Infinite sets	A set that has an unlimited number of elements is called an infinite set. For example, the set of all even counting numbers, {2, 4, 6, 8, 10, ...} is infinite. It does not stop. The set of all numbers that use the character 7 is infinite. It has elements such as 7, 17, 27, 70, 71, 234789, and 7827927 in it, along with an endless collection of other numbers. Strangely enough, there are very few infinite sets in the real world. Most of the inifinte sets that we will deal with are sets of numbers.