Linear Algebra, 2020a

A-6
P
x
The outer rectangle contains the universe Ω of all objects under discussion. For
instance, in a statement about real numbers, the rectangle encloses all members
of R. The set is pictured as a circle, enclosing its members.
Here is the diagram for P ⊆ Q. It shows that if x ∈ P then x ∈ Q.
P
Q
Set Operations The union oftwosetsisP ∪ Q = {x | (x ∈ P) or (x ∈ Q)}. The
diagram shows that an element is in the union if it is in either of the sets.
P
Q
The intersection is P ∩ Q = {x | (x ∈ P) and (x ∈ Q)}.
P
Q
The complement of a set P is P comp = {x ∈ Ω | x ∉ P }
P
x
Multisets As described above, a set is a collection in which order does not matter,
so that the sets {2, π} and {π, 2} are equal, and in which repeats collapse, so
that the sets {7, 7} and {7} are equal.