Introduction to Sets
Lecture 28
Section 6.1
Robb T. Koether
Hampden-Sydney College
Wed, Mar 5, 2014
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 1 / 36
1 Sets
2 Proving Set Relations
3 Set Operations
4 Power Sets
5 Cartesian Products
6 Assignment
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 2 / 36
Outline
1 Sets
2 Proving Set Relations
3 Set Operations
4 Power Sets
5 Cartesian Products
6 Assignment
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 3 / 36
Sets
Definition (Set)
A set is a collection of elements.
This “definition” does not really define what a set or an element is.
It merely substitutes the word “collection” for “set.”
But we already know what a set is, right?
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 4 / 36
Set Notation
Let S be a set and let P(x) be a predicate.
Then we can define a set A to be
{x ∈ S | P(x)}.
This means that A contains every element x of S for which P(x) is
true.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 5 / 36
Subsets
Definition (Subset)
A set A is a subset of a set B, denoted A ⊆ B, if x ∈ A→ x ∈ B.
Definition (Equality)
A set A is equal to a set B, denoted A = B, if x ∈ A↔ x ∈ B.
That is, A ⊆ B if every element of A is also an element of B.
And A = B if every element of A is an element of B and also every
element of B is an element of A.
Therefore, A = B if and only if A ⊆ B and B ⊆ A.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 6 / 36
Notation
If A ⊆ B, but A 6= B, then we write A ⊂ B and A is called a proper
subset of B.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 7 / 36
Outline
1 Sets
2 Proving Set Relations
3 Set Operations
4 Power Sets
5 Cartesian Products
6 Assignment
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 8 / 36
Proving the Subset Relation
To prove that A ⊆ B,
Let x be an arbitrary element (generic particular) of A.
That is, write “Let x ∈ A.”
Then show that x ∈ B.
Conclude that A ⊆ B.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 9 / 36
Proving Set Equality
To prove that A = B,
Prove first that A ⊆ B.
Then prove that B ⊆ A.
Conclude that A = B.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 10 / 36
Example
Theorem
Let
A = {n ∈ Z | n = 6k + 3 for some k ∈ Z}
and let
B = {n ∈ Z | n = 3k + 6 for some k ∈ Z}.
Then A ⊆ B.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 11 / 36
Example
Proof.
Let n ∈ A.
Then there exists k ∈ Z such that n = 6k + 3.
Let m = 2k − 1.
Then
3m + 6 = 3(2k − 1) + 6
= 6k + 3
= n.
So n ∈ B and therefore, A ⊆ B.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 12 / 36
Example
Theorem
Let
A = {n ∈ Z | n divides 8 and n divides 12}
and let
B = {n ∈ Z | n divides 4}.
Then A = B.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 13 / 36
Example
Proof.
(Proof that A ⊆ B)
Let n ∈ A.
Then n | 8 and n | 12.
So 8 = na and 12 = nb for some integers a and b.
It follows that 4 = 12− 8 = n(b − a).
So n | 4.
Therefore, n ∈ B and, thus, A ⊆ B.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 14 / 36
Example
Proof.
(Proof that B ⊆ A)
Let n ∈ B.
Then n | 4.
Because 4 | 8 and 4 | 12, it follows that n | 8 and n | 12.
Therefore, n ∈ A and, thus, B ⊆ A.
Therefore, A = B.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 15 / 36
Outline
1 Sets
2 Proving Set Relations
3 Set Operations
4 Power Sets
5 Cartesian Products
6 Assignment
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 16 / 36
Union and Intersection
Definition (Union)
The union of sets A and B, denoted A ∪ B, is the set
{x | x ∈ A or x ∈ B}.
Definition (Intersection)
The intersection of sets A and B, denoted A ∩ B, is the set
{x | x ∈ A and x ∈ B}.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 17 / 36
Union and Intersection
We may define the union and intersection of sets in terms of
predicates.
Let A = {x | P(x)} and let B = {x | Q(x)} for some predicates
P(x) and Q(x).
Then
A ∪ B = {x | P(x) ∨Q(x)}
and
A ∩ B = {x | P(x) ∧Q(x)}
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 18 / 36
The Universal Set
Definition
The universal set, denoted U, in any given situation is the set of all
elements under consideration.
Typically, the universal set will be Z or Q or R.
When the universal set is understood (or not relevant), we may
write simply
{x | P(x)}
rather than
{x ∈ S | P(x)}.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 19 / 36
Difference and Complement
Definition (Union)
The difference of set A minus set B, denoted A− B, is the set
{x | x ∈ A and x /∈ B}.
Definition (Complement)
The complement of a set A, denoted Ac , is the set
U − A = {x | x ∈ U and x /∈ A}.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 20 / 36
Difference and Complement
We may define the difference and complement of sets in terms of
predicates.
Let A = {x | P(x)} and let B = {x | Q(x)} for some predicates
P(x) and Q(x).
Then
A− B = {x | P(x) ∧ ∼ Q(x)}
and
Ac = {x | ∼ P(x)}
(Recall that the free variable x has a “domain” D.)
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 21 / 36
Disjoint Sets
Definition (Disjoint Sets)
Two sets A and B are disjoint if A ∩ B = ∅.
Definition (Mutually Disjoint Sets)
Set A1, A2, A3, . . . are mutually disjoint if Ai ∩ Aj = ∅ for all i and j
where i 6= j .
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 22 / 36
Example
Let E = {n ∈ Z | n is even}.
Then the odd integers are O = Z− E .
Furthermore, the sets E and O are disjoint.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 23 / 36
Example
Let
A0 = {n ∈ Z | n mod 3 = 0},
A1 = {n ∈ Z | n mod 3 = 1},
A2 = {n ∈ Z | n mod 3 = 2},
Then A1, A2, and A3 are mutually disjoint.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 24 / 36
Partition
Definition (Partition)
A partition of a set A is a collection of nonempty subsets
{A1, A2, A3, . . .} such that
A = A1 ∪ A2 ∪ A3 ∪ · · · and
A1, A2, A3, . . . are mutually disjoint.
In the last example, the collection of sets {A0, A1, A2} is a partition
of Z.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 25 / 36
Outline
1 Sets
2 Proving Set Relations
3 Set Operations
4 Power Sets
5 Cartesian Products
6 Assignment
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 26 / 36
Power Sets
Definition
The power set of a set A, denoted P(A), is the set of all subsets of A.
The power set of A includes A itself and the empty set.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 27 / 36
Example
Let A = {a, b, c}.
List the elements in P(A).
P(A) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
= {∅, {a}, {b}, {a, b}, {c}, {a, c}, {b, c}, {a, b, c}}.
What is P(∅)?
What is P(P(∅))?
What is P(P(P(∅)))?
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 28 / 36
The Power Set
Theorem
Let A be a set with n elements. Then P(A) contains 2n elements.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 29 / 36
The Power Set
Proof.
We proceed by induction on n.
When n = 0, let A be a set of 0 elements, i.e., A = ∅.
Also, P(A) = {∅}, which has 1 element, and 1 = 20.
So the statement is true when n = 0.
Robb T. Koether (Hampden-Sydney College) Introduction to Sets Wed, Mar 5, 2014 30 / 36
The Power Set
Proof.
Suppose the statement is true when n = k for some integer k ≥ 0.
Let A be a set with k + 1 elements.
Choose an element x ∈ A and let B = A− {x}.
Then B has k elements, so P(B) has 2k elements.
Then A ha
