Download Section 2.3Subsets

Section 2.3 Subsets
Definition of subset
We say that a set B is a subset of a set A, and write B ⊆ A, if every element of B is
also an element of A.
E.g., {1, 2} ⊆ {1, 2, 3}.
E.g., { odd numbers } ⊆ {natural numbers}
If B is not a subset of A, we write B 6⊆ A.
Another way to think of subsets: B ⊆ A if every element of B could be selected
from A.
Let B be a subset of A. We call B a proper subset of A if B is not all of A. In
symbols, B is a proper subset of A if B ⊆ A but B 6= A. If a subset is not proper,
then it is called improper. Of course, every set A has exactly one improper subset:
A itself.
E.g., {1, 2} is a proper subset of {1, 2, 3}; {1, 2, 3} is an improper subset of
{1, 2, 3} (and the only one).
A strange fact: the empty set is a subset of every set.
In symbols, 0/ ⊆ A for any set A (even if A is itself the empty set).
Examples
List all subsets of: 0/
List all subsets of: {1}
List all subsets of: {1, 2}
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List all subsets of: {1, 2, 3}
/
List all elements, and all subsets, of {0}
List all elements, and all subsets, of {α, β, {α, β}}.
Number of subsets
Theorem. A set with n elements has 2n subsets.
E.g., how many subsets has the set A = {Mary, Frank, Sarah}?
E.g., A sandwich can come with any combination of lettuce, tomato, and onion,
or with none of these. How many different combinations are possible?
E.g., A set A has 63 proper subsets. How many elements has A?
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Complement of a set
The complement of a set A is the set of all objects that are in the universe but not
in A. In symbols, A0 = {x | x 6∈ A}.
E.g., let A = {1, 2, 5}, and suppose the universe is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Then
A0 = {3, 4, 6, 7, 8, 9, 10}.
E.g., let A = {1, 2, 5}, and suppose the universe is all natural numbers. Then
A0 = {3, 4, 6, 7, 8, . . .}.
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