Sets

Summary

• A set is an unordered collection of distinct objects
• If \(A=\{ 1,2,3 \}\) then \(A\) is a set consisting of the object \(1,2\) and 3
• “Objects are also called elements or members of the set.
• If \(A=\{ 1,2,3 \}\) and \(B=\{ 3,2,1 \}\) then \(A=B\)
• If \(A=\{ 1,2,3 \}\) and \(B=\{ 1,1,2,2,3 \}\) then \(A=B\)
• \(∅\) denotes the empty set, \(∅=\{ \}\)
• A set may have an infinite number of elements, for example the set of natural numbers \(N\) .
• Elements of a set may be sets themselves
• \(a∈A\) means that \(a\) belongs to the set \(A\) . \(a∉A\) means that \(a\) does not belong to the set \(A\) .
• Set builder notation. Example: \(A=\{ x∈N | x<5 \}=\{ 1,2,3,4 \}\)
• Subsets: We write \(A⊆B\) and say that \(A\) is a subset of \(B\) if every element of \(A\) is also an element of \(B\) .
• Proper subsets: We write \(A⊊B\) and say that \(A\) is a proper subset of \(B\) if \(A⊆B\) and \(A≠B\) .
• The union of two sets \(A\) and \(B\) , denoted by \(A∪B\) , is the collection of elements that are in \(A\) or \(B\) (or both).
• The intersection of \(A\) and \(B\) , denoted by \(A∩B\) is the collection of elements that are in \(A\) and \(B\) .
• The set-theoretic difference of \(A\) and \(B\) , denoted by \(A∖B\) , is the collection of elements in \(A\) but not in \(B\) .
• If all sets under consideration are subsets of a given set \(Ω\) , then \(Ω\) is called the universal set .
• The complement of \(A\) , denoted by \(A^C\) or   \(\bar{A}\) or \(\tilde{A}\) or \(A'\) , is defined as the elements not in \(A\) , that is, \(A^C=Ω∖A\) .
• If \(A∩B =∅\) then we say that \(A\) and \(B\) are disjoint .
• A Venn diagram is a visual representation of one or more sets, often used to illustrate possible logical relations between the sets
• Elements are depicted as points in the plane
• Sets are depicted as regions inside closed curves
• The points inside the curve represent elements of the set.