1.1: Basics of Sets
An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a set .
A set is a collection of distinct objects. Individual objects in a set are called elements, or members, of the set.
For example, the name of each month is an element of the set of months in a calendar year. The elements of the set are listed inside a pair of braces, \(\{\text{ } \}\). The set of months in a calendar year can be written in set notation as follows:
A = { January, February, March, April, May, June, July, August, September, October, November, December }.
A set simply specifies the contents so the order in which elements are written makes no difference. Each element is listed only once. Thus, { March, April, May } and { May, April, March } are considered to be the same set. If two sets contain exactly the same elements, the sets are equal .
Commonly, capital letters are used to name sets to make it easier to refer to that set later.
We show that an element belongs to a set by using the symbol \(\in\), which means “is an element of.” The symbol \(\notin\) means "is not an element of."
A set that contains no elements, \(\{\text{ } \}\), is called the empty set and is notated \(\emptyset\).
Let \(A=\{1,2,3,4\}\).
To show that 2 is an element of the set \(A\), we'd write \(2 \in A\).
To show that 5 is not an element of the set \(A\), we'd write \(5 \notin A\).
Sets must be well-defined . That is, it must be very clear if an object is an element of the set or not.
For example, the set of letters in the word ghost is well-defined because it is clear that the letter g is an element of the set and the letter r is not an element of the set. However, the set of all diligent students in our class is not well-defined because diligence is matter of perception. Likewise, the set of all days last year with high temperatures below 32 degrees Fahrenheit is well-defined. However, the set of all cold days last year is not well-defined because is whether a day is cold not is subjective.
Sets can be described in three common ways: using a verbal description, listing its elements in roster form, and using set builder notation as seen in these examples.
1. Sets can be defined by describing its contents in words. For example,
- The set of all even numbers
- The set of all books written about travel to Chile
2. Sets can be defined by listing the elements in roster form using set braces. For example,
- \(\{1, 3, 9, 12\}\)
- \(\{\text{red, orange, yellow, green, blue, indigo, purple}\}\)
- \(\{1, 4, 9, 16, 25, 36, 49, ... \}\) This is an example of an infinite set . The ellipsis (...) indicates that the set continues in this pattern and there is no end.
3. Sets can be defined using set builder notation. For example,
- \(B = \{x | x \in N \ \text{ and }1<x \leq8\}\) means Set \(B\) is the set of natural numbers greater than 1 but less than or equal to 8, or \(B=\{2,3,4,5,6,7,8\}\).
- \(C = \{2x+1\ \text{ where } x =3, 4, 5\}\) means \(C=\{7,9,11\}\).
Let \(D = \{n | n \in N \ \text{ and } n \text{ is an odd number}\}\). Write set \(D\) in roster notation and using a verbal description.
- Answer
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\(D=\{1,3,5,7,9,11, ... \}\). Set \(D\) is the set of all odd numbers. This is another example of an infinite set.
Sometimes it is important to know how many elements are in a set. The cardinality of a set \(S\), denoted \(n(S)\), indicates the number of elements in set \(S\). If \(S=\{a, b\}\), the cardinality of set \(S\) is 2, and this is written \(n(S) = 2\).
The number of elements in a set is the cardinality of that set.
The cardinality of the set \(A\) is often notated \(n(A)\).
What is the cardinality of \(P,\) the set of English names for the months of the year?
Solution
The cardinality of this set is \(12,\) since there are 12 months in the year. We can write \(n(P)=12\).
Two sets are said to be equivalent set if they have both have the same cardinality. That is, set \(A\) is equivalent to set \(B\) if \(n(A)\) = \(n(B)\). We write \(A \cong B\) to show these two sets are equivalent.
The term equivalent should not be confused with equal . Remember that two sets are equal when they contain exactly the same elements. The difference between the terms equal and equivalent is demonstrated in the next example.
Consider the four sets: \(A=\{p,q,r,s\} \quad B=\{a,b,c\} \quad C=\{x,y,z\} \quad D=\{b,a,c\}\).
Compare the sets using the terms equal and equivalent .
Solution
- Sets \(A\) and \(B\): These sets are not equivalent (\(A \ncong B\)) because they have different cardinality. They are not equal (\(A \neq B\)) because they do not contain exactly the same elements.
- Sets \(B\) and \(C\): These sets are equivalent (\(B \cong C\)) because they both have cardinality 3. They are not equal (\(B \neq C\)) because they do not contain exactly the same elements.
- Sets \(B\) and \(D\): These sets are equivalent (\(B \cong D\)) because they both have cardinality 3. They are equal (\(B = D\)) because they contain exactly the same elements.
- Sets \(C\) and \(D\): These are equivalent (\(C \cong D\)) because they both have cardinality 3. They are not equal (\(C \neq D\)) because they do not contain exactly the same elements.
Consider the five sets:
\(P=\{a,b,c,d\} \quad Q=\{x,y,z,w\} \quad R=\{c,d,a,b\} \quad S=\{z,y,x\} \quad T=\{x | x \in N \ \text{ and }1\leq x \leq4\}\)
- Which sets are equal?
- Which sets are equivalent?
- Answer
-
- \(P = R\)
- \(P \cong Q\), \(P \cong R\), \(P \cong T\), \(Q \cong R\), and \(Q \cong T\)
Sometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. While Chris’s collection is a set, we can also say it is a subset of the larger set of all Madonna albums.
A subset of a set \(A\) is another set that contains only elements from the set \(A\), but may not contain all the elements of \(A\).
If \(B\) is a subset of \(A,\) we write \(B \subseteq A\).
More formally,
- If every element of set \(B\) is also an element of set \(A\), then \(B \subseteq A\).
- If \(B \subseteq A\), then from \(x \in B\) it follows that \(x \in A\).
A proper subset is a subset that is not identical to the original set.
If \(B\) is a proper subset of \(A\), we write \(B \subset A\).
Consider these three sets: \(A=\) the set of all even numbers\(\quad B=\{2,4,6\} \quad C=\{2,3,4,6\}\).
Here, \(B \subset A\). Every element of \(B\) is also an even number and is also an element of \(A\). However, there are elements in set \(A\) that are not in set \(B\).
It is also true that \(B \subset C\).
\(C\) is not a subset of \(A\), since \(C\) contains an element, 3, that is not contained in \(A\).
Suppose a set contains the plays “Much Ado About Nothing”, “MacBeth”, and “A Midsummer’s Night Dream”. What is a larger set this might be a subset of?
Solution
There are many possible answers here. One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all works of British literature.
Try it Now 3
The set \(A=\{1,3,5\} .\) What is a larger set this might be a subset of?
- Answer
-
There are several answers. For example,
- The set of all odd numbers less than 10.
- The set of all odd numbers.
- The set of all integers.
- The set of all real numbers.
Sometimes it is useful to list all the subsets of a set. In other instances, it is important to know how many subsets there will be without listing them all. To develop a formula for the number of subsets of a set, it will be helpful to consider a few examples and look for a pattern.
- A set with no elements : The empty set \(\{\text{ } \}\) has only one subset -- itself.
- A set with one element: Suppose we add one more element, \(a\), to the empty set to form the set \(\{a\}\). This new set still has the subset \(\{\text{ } \}\) from the previous set. However, there is also a new subset, \(\{a\}\). So, a set with one element, has two subsets.
- A set with two elements: Suppose we add one more element, \(b\), to the previous set to form the set \(\{a,b\}\). This new set has subsets \(\{\text{ } \}\), \(\{a\}\), \(\{b\}\), and \(\{a,b\}\). So, a set with two elements, has four subsets.
- A set with three elements: Suppose we add one more element, \(c\), to the previous set to form the set \(\{a,b,c\}\). This new set has subsets \(\{\text{ } \}\), \(\{a\}\), \(\{b\}\), \(\{c\}\), \(\{a,b\}\), \(\{a,c\}\), \(\{b, c\}\), and \(\{a,b,c\}\). So, a set with three elements, has eight subsets.
The table summarizes the number of subsets that can be formed based on the cardinality of the set.
| Number of elements in set | Number of subsets |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
Notice that each time an extra element is included in a set, its number of subsets double. Since the empty set has one subset and each additional element doubles the number of subsets, a set with \(n\) elements has \(2^{n}\) subsets.
If set \(A\) has \(n\) elements, then set \(A\) has \(2^{n}\) subsets.
Suppose the set \(V\) is the set of all vowels in the English alphabet: \(V=\{a, e, i, o, u\}\).
How many subsets can be formed using the elements of set \(V\)?
Solution
Set \(V\) contains 5 elements. The number of subsets that can be formed can be found using \(2^{n}=2^{5}=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2=32.\)