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Proper Subset:
A proper subset
is a special type of subset.
There are two requirements for set
A to be a proper
subset of set B.
They are:
1.
A is a subset of
B,
i.e.,
A
⊆
B
and
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 since not every member of the U.S. Senate is on the
Judiciary Committee.
Example:
For A = {red,
blue} and B = {red,
white, blue}, A
⊂
B since both
requirements are met:
1.
A
⊆
B since red and blue
are in both sets A
and B;
and
2.
A ≠
B
since set B
contains the element “white” but set
A does not.
Example:
The set {a, b, c} has
7 proper subsets.
They are:
∅,
{a}, {b},
{c}, {a,
b}, {a, c}, and {b,
c}.
Note that {a, b, c}
is not a proper subset of {a, b, c}. Also, note that there is
always one less proper subset than there are subsets of a set
since a set cannot be a proper subset of itself.