Elements

An element of a set \(A\) is something that belongs to \(A.\) The notation \(x \in A\) mean \(x\) is an element of \(A\). The notation \(x \notin A\) means \(x\) is not an element of \(A\).

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Consider the following sets: \[A = \{1, 2\}, B=\{1, 3\}\] Then \(1 \in A\), \(1 \in B\), \(2 \in A\), \(2 \not\in B\), \(3 \not\in A\) and \(3 \in B\).

Subsets

Definition: If every element of a set \(A\) belongs to a set \(B,\) we say \(A\) is a A subset of \(B.\) The notation \(A \subset B\) means \(A\) is a subset of \(B\). The notation \(A \not\subset B\) means \(A\) is not a subset of \(B\).

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Consider the following sets: \[A = \{1, 2\}, B=\{1, 3\}, C = \{1, 2, 3\}, D = \{1, 3, 4\}\] Then \(A \subset C\), \(B \subset C\), \(B \subset D\) and \(B \subset B\). However, \(A \not\subset B\), \(B \not\subset A\) and \(C \not\subset A\).

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The set \(B\) is completely contained in \(A,\) so \(B \subset A\).

\(A = \{1, 5, 7\}\) and \(B = \{5, 7\}\)
Element of \(B\) is also an element of \(A\).