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.
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.
\(\mathtt{y}\notin \{\mathtt{a},\mathtt{e},\mathtt{i},\mathtt{o},\mathtt{u}\}\) 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 \(\mathtt{u}\text{.}\)”
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.
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.
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.
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 \(\{2\}\in\{\{1\},\{2\},\{3,4\}\}\text{.}\)
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.
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 \(A\neq B\text{.}\)
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.
We give examples of the correct usage of the symbols \(=\) and \(\neq\text{.}\)
\(\{1,2,3\} = \{2,1,3\}\text{,}\) 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.
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.
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.
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.
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 \(C\text{.}\)
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 \(\{1,3,5,6\}\text{.}\)
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.