Notes by Lex Toumbourou

1.104 The definition of a set

• Set Theory
  • a branch of maths about well-defined collections of objects.
  • concept of sets by Georg Cantor, a German mathematician.
  • forms the basis of other fields:
    • counting theory
    • relations
    • graph theory
    • finite state machines
• Set
  • a collection of any kind of "well-defined" objects:
    • people, ideas, numbers etc.
  • Set is unordered and contains unique objects.
  • Examples and notation:
    • Set of positive even integers < 10: $E = \{2, 6, 4, 8\}$
    • Set of vowels in English alphabet: $V = \{a, e, i, o, u\}$
    • Set of colours: $C = \{red, green, blue\}$
    • Empty set (a set containing nothing): $\{\}$ = $\emptyset$
  • In math notation, use $\in$ to represent that something is element of set: * $2 \in E$.
  • Use $\notin$ if not an element of set:
    • $3 \notin E$.
• Cardinality
  • Given set $S$, the cardinality of $S$ is the number of elements contained in $S$.
  • Write cardinality as $|S|$
  • Example: $|C| = 3$
• Subset
  • Express as: $\subseteq$
    • Latex: \subseteq
  • $A$ is a subset of $B$ if and only if every element of $A$ is also in $B$.
    • $A \subseteq B$.
  • This gives us equivalence:
    • $A \subseteq B \iff x \in A \text{ then } x \in B \text{(for all x)}$
  • Any set is a subset of itself: $S \subseteq S$
• Empty Set
  • is a subset of any set $\emptyset \subseteq S$
  • empty set is a subset of itself: $\emptyset \subseteq \emptyset$
• Special Sets
  • $\mathbf{N}$ = set of natural numbers = $\{1, 2, 3, 4, ...\}$
  • $\mathbf{Z}$ = set of integers = $\{..., -3, -2, -1, 0, 1, ...\}$
  • $\mathbf{Q}$ = set of rational numbers (of form a/b where a and b are elements of Z and b $\ne$ 0)
  • $\mathbf{R}$ = set of real numbers
  • $\mathbf{N} \subseteq \mathbf{Z} \subseteq \mathbf{Q} \subseteq \mathbf{R}$

1.106 The listing method and rule of inclusion

• Set Representation Methods
  • Listing Method
  • Set Builder Notation
• Listing Method
  • Represent a set $S$ using all elements of set $S$.
  • Examples:
    • Set of all vowels in English alphabet.
      • $S1 = \{a, e, i, o, u\}$
    • Set of all positive integers less than 10.
      • $S2 = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$
• Set Builder Notation
  • Examples:
    • Set of all even integers: { ..., -6, -4, -2, 0, 2, 4, 6 ... }
      • $\text{Even} = \{2n | n \in Z \}$
      • $\text{Odd} = \{2n+1 | n \in Z \}$
    • Set of rational numbers $Q$: { ..., 1/1, 1/2, 1/3, 1/4, ... }
      • $Q = \{ \frac{n}{m} | n,m \in \text{Z and m }\neq 0 \}$
    • Set of things in my bag: {pen, book, laptop, ...}
      • $\text{Bag =} \{x|x \text{ is in my bag } \}$
  • Exercise
    • Rewrite the following sets using listing method
      • $S_1 = \{ 3n | n \in N \text{ and } n < 6\}$
        • $S_1 = {3, 6, 9, 12, 15}$
      • $S_2 = \{2^n|n \in Z \text{ and } 0 \leq n \leq 4 \}$
        • $S_2 = {1, 2, 4, 8, 16}$
      • $S_3 = \{2^{-n}|n \in Z \text{ and } 0 \leq n \leq 4 \}$
        • $S_3 = \{1,\frac{1}{2}, \frac{1}{4}, ... \}$
    • Rewrite the following sets using set building method
      • $S_1 = \{ \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, ... \}$
        • $S_1 = {\frac{1}{2n} | n \in \text{Z and } 0 < n \leq 5}$

1.108 The Powerset of a set

• Powerset
  • A set can have another set as its element. * $\{5, 6\} \in \{\{5, 6\}, \{7, 8\}\}$ * $\{5,6\} \subseteq \{5, 6, 7\}$
  • A set containing all subsets of another set.
  • The powerset of $S$ is $P(S)$ which is the set containing all subsets of $S$.
  • Example
    • $S = \{1, 2, 3\}$
    • $P(S) = \{ \emptyset, \{1\}, \{2\},\{3\},\{1, 2\},\{1, 3\}, \{2, 3\}, \{1, 2, 3\} \}$
  • Exercise 1
    • $S = \{ a, b \}$
    • $P(S) = \{\emptyset, \{a\}, \{b\}, \{a, b\} \}$
    • $\{a\} \in P(S)$
    • $\emptyset \subseteq P(S)$ - empty set is subset of $P(S)$
    • $\emptyset \in P(S)$ - empty set is also in $P(S)$
  • Exercise 2
    • $P(\emptyset) = ?$
    • $P(P(\emptyset)) = ?$
    • Powerset of empty set is set containing empty set: $P(\emptyset) = \{ \emptyset \}$
    • Empty set is the only subset of the empty set: $\emptyset \subseteq \emptyset$
    • Empty set is a set subset of the power set of empty set: $\emptyset \subseteq P(\emptyset)$
    • $P(P(\emptyset)) = \{ \emptyset, \{ \emptyset \} \}$
  • Cardinality of a powerset
    • $|P(S)| = 2^{|S|}$
      • $S = \{1, 2, 3\}$, $|S| = 3$, $|P(S)| = 2^3 = 6$
      • $P(S) = \{ \emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{2, 3\} \}$

1.110 Set operations

• Union

  • The union of two sets are all element in either A or B.
  • Notion: $A \cup B$
  • Latex operator \cup
    • "The U thing"
  • Set builder: $A \bigcup B = \{ x | x \in A \text{ or } x \in B \}$

  • Python example:

    A = {1, 2}
    B = {2, 3}
    A.union(B)
    # {1, 2, 3}
    

  • Think of a union between people: it's when people come together to make up a larger set.

  • Membership table
    • Show all combinations of sets an element can belong to.
      • Put 1 if element belongs to set.
      • Put 0 if it doesn't

$A$	$B$	$A \cup B$
0	0	0
0	1	1
1	0	1
1	1	1

• Intersection
  • Notion: $A \cap B$
  • Set builder: $A \cap B = \{ x | x \in A \text{ and } x \in B \}$
  • Looks like a horse shoe. The kind you'd take to a dirt road intersection.
  • Think of an intersection between roads: it's the part of road that both of them share.
  • Membership table

$A$	$B$	$A \cup B$
0	0	0
0	1	0
1	0	0
1	1	1

• Set Difference
  • Elements in $A$, but not $B$.
    • $A - B = \{ x | x \in A \text{ and } x \notin B \}$
  • Example:
    • $\{1, 2\} - \{2, 3\} = \{1\}$
  • Membership table:

$A$	$B$	$A \cup B$
0	0	0
0	1	0
1	0	1
1	1	0

• Symmetric Difference
  • Elements in $A$ or in $B$ but not in both.
    • $A \oplus B = \{ x | (x \in A \text{ or } x \in B) \text{ and } x \notin A \cap B \}$
  • Latex: \oplus
  • Can think of it as union of A and B, with all the common elements of A and B removed.
    • $A \oplus B = (A \cup B) - (A \cap B)$
  • Example:
    • A = {1, 2, 3}
    • B = {3, 4, 5}
    • $A \oplus B = \{ 1, 2, 4, 5 \}$
  • Membership table

$A$	$B$	$A \cup B$
0	0	0
0	1	1
1	0	1
1	1	0

1.112 Essential reading

• Koshy, Thomas. Discrete Mathematics with Applications:
  • pp. 67-70 and pp. 72- 75
  • pp.76: Exercises: 1–8, 13-27, 30-32 and 41-44.