Subsets

1
Section 2.2

• Subsets

2
Subsets

• Set A is a SUBSET of Set B if all the elements of
  A are contained in B.
• Write A B

3
Examples of Subsets

• Example
• A 5,6,7,8 and B 6,7
• Is B a subset of A?
• Example
• A x x is a state east of the Mississippi
  River and
• B x x is a state bordering the Atlantic
  Ocean
• Is B a subset of A?

4
Examples of Subsets

• Example
• A is the set of states. B is the set of state
  capitals.
• Is B a subset of A?
• Example
• A c, w, r. B w, r, c.
• Is B a subset of A?
• If B A then B A .

5
Proper Subset

• B is a proper subset of A if B is a subset of A
  and there is at least one element of A that is
  not in B.
• Write B A
• Example
• A is the set of toys requiring batteries.
• B is the set of toys requiring AA batteries.
• Which of the following is true?
• A B B A
• A B B A

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Subsets Proper Subsets

• Every set is a subset of itself.
• No set is a proper subset of itself.
• The empty set is a subset of every set.

7
Element vs. Subset

• Know when an object is an element or a subset.
• A B means that A and B are both sets and A
  is a subset of B.
• x ? B means that x is an element of B, but not a
  subset of B. An element cannot be a subset.

8
Number of Subsets

• How many different subsets can be made from the
  set
• A red,blue
• 4 altogether
• red,blue
• red
• blue

9
Number of Subsets

• What is the formula for the number of subsets of
  a set?
• ( Hint Focus on the cardinal number, n, of the
  set.)
• The formula is 2n, where n is the number of
  elements in the set.

10
Number of Subsets

• How many different subsets can be made from
• A red, blue, green
• The formula is 23 8, since there are 3 elements
  in the set. We can now list all the subsets.

11
Subsets

• We know there are 8 subsets, lets list them. To
  make our work easier, we will abbreviate r,b,g
  for
• red, blue, green
• r,b,g, , r, b,g,r,b,
• r,g,b,g

12
Proper Subsets

• All of the sets are proper subsets except the set
  itself r, b, g .
• There are 7 proper subsets
• The formula for the number of proper subsets is
• 2n -1 23 1 8 -1 7