**
Readings for Session 4
**

**
Subsets of a Set**

**
***W**hat
is the relationship between the U.S. Senate and its Judiciary
Committee?*

*
*

*
In
2009, the two senators from Minnesota, Amy Klobucher and Al
Franken were both members of the U.S. Senate’s Judiciary
Committee, but Minnesota’s 7 ^{th} Congressional District
representative, Collin Peterson, could not be a member of that
committee. *

In order to be a member of the
Judiciary Committee a person must first be a member of the U.S.
Senate. This is an example of a collection of objects chosen
from another collection. This leads to the concept of a subset.*
***
**

**
Subset:
**A set

*Notation:*
*A*
⊆
*B*
is read, “Set *A*
is a subset of set
*B.*”

*Example:* In
the paragraph above, 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.

** Listing
Subsets:
**List
all the subsets of {

*Example:*
The set {*a, b, c*} has
8 subsets. They
are:

∅,
{*a*}, {*b*},
{*c*}, {*a,
b*}, {*a, c*}, {*b,
c*}, and {*a, b, c*}.

**
Proper Subset:**
A

1.
*A* is a subset of
*
B,
i.e.,
**
A*
⊆
*B*
*
and*

Example:

1.

** Listing
Proper Subsets:
**List
all the proper subsets of {

*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.

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