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Subset: A set A is a subset of a set B if every element of A is also an element of B.
Notation:
A
⊆
B
is read, “Set A
is a subset of set
B.”
Example: The set of members of the U.S. Senate’s
Judiciary Committee is a subset of the set of members of the
U.S. Senate.
Example:
For A = {red, blue}
and B = {red, white,
blue},
A
⊆
B since every element
of A is also an
element of
B.
Example: Let C
= {a, b, c} and
D = {b,
c, d, e}. Then
C
⊈
D, read “C
is not a subset of D”
since a is an element
of C but not an
element of
D. Also,
D
⊈
C since
e
∈
D and
e
∉
C.
Example:
A
⊆
A and
∅
⊆
A.
A set is a subset of itself since a set contains all its
elements. Also, the empty set is a subset of every set, because
every element in the empty set belongs to any set since the
empty set has no elements.
Example:
The set {a, b, c} has
8 subsets. They
are:
∅,
{a}, {b},
{c}, {a,
b}, {a, c}, {b,
c}, and {a, b, c}.