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Integers and Modern Applications for the Uninitiated

Sebastian Pauli

Section 5.3 Membership and Equality

The basic relationship between a set and an object is whether or not the object is an element of the set. We can only ask whether an element is in a set or not. There is also no ordering of the elements in the set.
In the video in Figure 5.13 we introduce the notation for is element of, define what it means for two sets to be equal, and give examples.
Figure 5.13. Membership and Equality by Matt Farmer and Stephen Steward

Definition 5.14.

The symbol \(\in\text{,}\) read as “is an element of” or “is in,” indicates membership in a set. The symbol \(\notin\text{,}\) read as “is not an element of” or “is not in,” indicates lack of membership in a set.
We show how to read the symbols \(\in\) and \(\not\in\text{.}\)

Example 5.15. \(\in\) and \(\not\in\).

  1. \(3 \in \{1, 2, 3, 4\}\) is read as “3 is an element of the set containing 1, 2, 3, and 4” or “3 is in the set containing 1, 2, 3, and 4.”
    This statement is true, as 3 is listed in \(\{1, 2, 3, 4\}\)
  2. \(3 \notin \{1, 2, 3, 4\}\) is read as “3 is not an element of the set containing 1, 2, 3, and 4” or “3 is not in the set containing 1, 2, 3, and 4.”
    This statement is false, as 3 is listed in \(\{1, 2, 3, 4\}\)
  3. \(5 \notin \{1, 2, 3, 4\}\) is read as “5 is not an element of the set \(\{ 1, 2, 3, 4 \}\)” or “5 is not in the set \(\{ 1, 2, 3, 4 \}\text{.}\)”
    This statement is true, as \(5\) is not in the listed in \(\{1, 2, 3, 4\}\text{.}\)
  4. \(5 \in \{1, 2, 3, 4\}\) is read as “5 is an element of the set \(\{ 1, 2, 3, 4 \}\)” or “5 is in the set \(\{ 1, 2, 3, 4 \}\text{.}\)”
    This statement is false, as \(5\) is not in the listed in \(\{1, 2, 3, 4\}\text{.}\)
  5. \(\mathtt{y}\notin \{\mathtt{a},\mathtt{e},\mathtt{i},\mathtt{o},\mathtt{u}\}\) is read as “\(\mathtt{y}\) is not an element of the set containing \(\mathtt{a}\text{,}\)\(\mathtt{e}\text{,}\)\(\mathtt{i}\text{,}\)\(\mathtt{o}\text{,}\) and \(\mathtt{u}\text{.}\)”
    This statement is true, because \(\mathtt{y}\) is not listed in \(\{\mathtt{a},\mathtt{e},\mathtt{i},\mathtt{o},\mathtt{u}\}\text{.}\)
In Checkpoint 5.16 decide whether statements with \(\in\) and \(\not\in\) are true or false.

Checkpoint 5.16. \(a \in S\) or \(a \not\in S\).

Let \(S=\lbrace -4,-3, -2,\ldots,17 \rbrace\text{.}\) For each statement indicate whether it is true or false.
  1. \(\displaystyle -5\not\in S\)
  2. \(\displaystyle 24\not\in S\)
  3. \(\displaystyle 20\in S\)
  4. \(\displaystyle 7\in S\)
Sets are collections of objects. In many of our examples these objects are number, but we have also encountered sets containing letters or colors. Sets themselves are also mathematical objects and thus can be contained in sets. In Problem 5.17 we consider statements that involve sets that contain sets.

Problem 5.17. \(\in\) and sets containing sets.

Decide whether the following statements are true or false.
  1. \(\displaystyle 2\in\{1,2,3,4\}\)
  2. \(\displaystyle \{2\}\in\{1,2,3,4\}\)
  3. \(\displaystyle \{2\}\in\{\{1\},\{2\},\{3,4\}\}\)
  4. \(\displaystyle 2\in\{\{1\},\{2\},\{3,4\}\}\)
Solution.
  1. The statement \(2\in\{1,2,3,4\}\) is read the integer \(2\) is in the set containing the integers \(1\text{,}\) \(2\text{,}\) \(3\text{,}\) and \(4\text{.}\) As \(2\) is in this list of elements in \(\{1,2,3,4\}\text{,}\) the statement \(2\in\{1,2,3,4\}\) is true.
  2. The statement \(\{2\}\in\{1,2,3,4\}\) means \(\{2\}\) is in the set containing \(1\text{,}\) \(2\text{,}\) \(3\text{,}\) and \(4\text{.}\) As \(\{2\}\) is not in this list of elements, the statement \(\{2\}\in\{1,2,3,4\}\) is false.
  3. The statement \(\{2\}\in\{\{1\},\{2\},\{3,4\}\}\) means \(\{2\}\) is in the set containing \(\{1\}\text{,}\) \(\{2\}\text{,}\) and \(\{3,4\}\text{.}\) As \(\{2\}\) is in this list of element, the statement \(\{2\}\in\{\{1\},\{2\},\{3,4\}\}\text{.}\)
  4. The statement \(2\in\{\{1\},\{2\},\{3,4\}\}\) is read the integer \(2\) is in the set containing \(\{1\}\text{,}\) \(\{2\}\text{,}\) and \(\{3,4\}\text{.}\) As \(2\) is not equal to any of \(\{1\}\text{,}\) \(\{2\}\text{,}\) and \(\{3,4\}\text{,}\) the statement \(2\in\{\{1\},\{2\},\{3,4\}\}\) is false.
We employ the notion of element being in a set in our definition of equality of sets.

Definition 5.18.

Two sets \(A\) and \(B\) are equal if each element in \(A\) is in \(B\) and if each element in \(B\) is in \(A\text{.}\) If two sets \(A\) and \(B\) are equal, we write \(A=B\text{.}\) If two sets \(A\) and \(B\) are not equal, we write \(A\neq B\text{.}\)
Verify that you have understood the definitions in this section by completing them in Checkpoint 5.19.

Checkpoint 5.19. Definition of \(\in\) and \(\not\in\) and \(=\) and \(\neq\) for sets.

Complete the following:
Let \(A\) and \(B\) be sets.
When \(b\)
  • select
  • is related to
  • is an element of
  • is equal to
  • is not related to
  • is not an element of
  • is not equal to
the set \(A\) we write \(b \in A\text{.}\)
When \(b\)
  • select
  • is related to
  • is an element of
  • is equal to
  • is not related to
  • is not an element of
  • is not equal to
the set \(A\) we write \(b\not\in A\text{.}\)
We say the set \(A\)
  • select
  • is related to
  • is an element of
  • is equal to
  • is not related to
  • is not an element of
  • is not equal to
the set \(B\) and write \(A=B\) if each element in the set \(A\)
  • select
  • is related to
  • is an element of
  • is equal to
  • is not related to
  • is not an element of
  • is not equal to
the set \(B\) and if each element in the set \(B\)
  • select
  • is related to
  • is an element of
  • is equal to
  • is not related to
  • is not an element of
  • is not equal to
the set \(A\text{.}\)
If the set \(A\)
  • select
  • is related to
  • is an element of
  • is equal to
  • is not related to
  • is not an element of
  • is not equal to
the set \(B\) we write \(A\ne B\text{.}\)
To prove that two sets are not equal we only need to find one element that is in one of the sets but not in the other set. To prove that two sets are equal we need to check all elements in both sets.

Example 5.20. Equality.

We give examples of the correct usage of the symbols \(=\) and \(\neq\text{.}\)
  1. \(\{1,2,3\} = \{2,1,3\}\text{,}\) as each element, namely 1, 2, and 3, of \(\{1,2,3\}\) is in \(\{2,1,3\}\) and vice versa. The order in which the elements are listed in roster form does not change the set.
  2. \(\{1,2\} \neq \{1,2,3\}\text{,}\) as 3 is not in \(\{1,2\}\text{.}\)
  3. \(\{a,b,c\}\neq \{1,2,3\}\text{,}\) as \(a\) is not in \(\{1,2,3\}\text{..}\)
  4. \(\{\}\neq \{1\}\) as the number 1 is not contained in the empty set.

Problem 5.21. Equal or not equal.

Let \(C:=\{1,3,5,6\}\text{.}\) For each statement indicate whether it is true or false.
  1. \(\displaystyle \{3\}=C\)
  2. \(\displaystyle C=\{6\}\)
  3. \(\displaystyle \{5,3,1,6\}=C\)
  4. \(\displaystyle \{5\}\ne C\)
Solution.
Recall that two sets are equal if all elements in the first set are in the second set and if all elements of the second set are in the first set.
  1. The number \(1\) is in the set \(C\) on the right but not in the set \(\{3\}\) on the left. So \(\{3\}\) is not equal to \(C\text{;}\) the statement is false.
  2. The number \(1\) is in the set \(C\) on the left but not in the set \(\{6\}\) on the right. So \(C\) is not equal to \(\{6\}\text{;}\) the statement is false.
  3. We first check whether every element of the set \(\{5,3,1,6\}\) is in the set \(C\) on the right.
    The number \(5\) is in the set \(C\text{.}\) The number \(3\) is in the set \(C\text{.}\) The number \(1\) is in the set \(C\text{.}\) The number \(6\) is in the set \(C\text{.}\)
    Now we are halfway done. Next we check whether every element of \(C=\{1,3,5,6\}\) is in \(\{5,3,1,6\}\text{.}\)
    The number \(1\) is in the set \(\{1,3,5,6\}\text{.}\) The number \(3\) is in the set \(\{1,3,5,6\}\text{.}\) The number \(5\) is in the set \(\{1,3,5,6\}\text{.}\) The number \(6\) is in the set \(\{1,3,5,6\}\text{.}\)
    We conclude that \(\{5,3,1,6\}=C\text{.}\) So the statement is true.
  4. The number \(1\) is in the set \(C\) on the right but not in the set \(\{6\}\) on the left. So \(\{5\}\) is not equal to \(C\text{,}\) in symbols: \(\{5\}\ne C\text{.}\) The statement is true.
In Checkpoint 5.22 you are asked to give a reason when you decide that the two sets are not equal.

Checkpoint 5.22. Equality of sets.

For the given sets \(C\) and \(D\) determine whether the statement
\begin{equation*} C=D \end{equation*}
is true or false. If the statement is false choose the reason.
  1. when \(C:=\lbrace 5\rbrace\) and \(D: = \lbrace \rbrace\)
  2. when \(C:=\lbrace 4, 1,0,2\rbrace\) and \(D: = \lbrace 2, 1, 0 \rbrace\)
  3. when \(C:=\lbrace 1, 2, 3\rbrace\) and \(D: = \lbrace 0, 1, 2, 3 \rbrace\)
  4. when \(C:=\lbrace 5,2, 1,0,4\rbrace\) and \(D: = \lbrace 2, 1, 0, 4, 5 \rbrace\)
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