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mathematicsprinted 11 · PDF 12
165 pagesrevision 46935
printed 11 · PDF 12
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If A ⊂ B, then A ⊆ B, but if A ⊆ B, then A does not necessarily have to be a
proper subset of B
A ∩ B represents the intersection of A and B. The intersection of two sets
refers to the elements that are common among both sets. E.g. if A = {1, 2, 3},
and B = {2, 4, 6}, then A ∩ B = {2}
A ∪ B represents the union of A and B. The union of two sets refers to all of
the elements in either or both sets. E.g. if A = {1, 4, 9} and B = {1, 3, 5}, then A
∪ B = {1, 3, 4, 5, 9}
A′ represents the complement of A. This refers to all the elements excluding
those in set A. E.g. if ε denotes {1, 2, 3, …, 20}, and A = {1, 4, 9, 16}, then A′ =
{2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20}
In any case, ε′ = ∅, as the complement of ALL the elements is taken, which
leaves no elements
Venn Diagrams
A Venn diagram is a way to illustrate all the elements within sets, as well as
any intersections amongst them
Venn diagrams usually have a rectangle to illustrate ε, and circles to represent
smaller sets. These circles may overlap depending on whether the sets have
any common elements. All circles are entirely inside the rectangle
In the above Venn diagram, ε is represented by the entire rectangle, set A is
represented by the red and pink area, and set B is represented by the blue
and pink area