Number of relations

Let set A: $A = \{\ \ 1, \ \ 2\ \ \}$

Powerset of A: $P(A) = \{\ \ \varnothing,\ \ \{1\},\ \ \{2\},\ \ \{1,2 \}\ \ \}$

Cartesian product of the set $A$: $A\times A = \{ \ \ (1,1),\ \ (1,2),\ \ (2,1),\ \ (2,2)\ \ \}$

Enumerated powerset of the Cartesian product of the set $A$ i.e. $\mathcal{P}(A\times A) = \mathcal{P}(A^2)$ is given below. There are 16 elements in this powerset i.e. there are 16 possible relations:

• 1 possible way to choose no pairs (0-place): $4\choose 0$

• 4 possible ways to choose 1 pair (1-place): $4\choose 1$

• 6 possible ways to choose 2 pairs (2-place): $4\choose 2$

• 4 possible ways to choose 3 pairs (3-place): $4\choose 3$

• 1 possible way to choose all pairs (n-place): $4\choose 4$

The formula to choose k-places out of n elements:

$${n \choose k} = {\frac{n!}{k!(n-k)!}}$$

Enumerated powerset of the Cartesian product of the set $A$:

$${\\ \mathcal{P}(\ \ \{\ \ (1,1),\ \ (1,2),\ \ (2,1),\ \ (2,2)\ \ \}\ \ ) = \\ \Huge\{\\ R_1: \varnothing, \\ R_2: \{(1,1)\}, \\ R_3: \{(1,2)\}, \\ R_4: \{(2,1)\}, \\ R_5: \{(2,2)\}, \\ R_6: \{(1,1), (1,2)\}, \\ R_7: \{(1,1), (2,1)\}, \\ R_8: \{(1,1), (2,2)\}, \\ R_9: \{(1,2), (2,1)\}, \\ R_{10}: \{(1,2), (2,2)\}, \\ R_{11}: \{(2,1), (2,2)\}, \\ R_{12}: \{(1,1), (1,2), (2,1)\}, \\ R_{13}: \{(1,1), (1,2), (2,2)\}, \\ R_{14}: \{(1,1), (2,1), (2,2)\}, \\ R_{15}: \{(1,2), (2,1), (2,2)\}, \\ R_{16}: \{(1,1), (1,2), (2,1), (2,2)\} \\ \Huge\} \\}$$

Relations:

• null (1): $R_1$
• identity (1): $R_8$
• universal (1): $R_{16}$
• reflaxive (4): $R_1, R_8, R_{13}, R_{16}$
• irreflexive (4): $R_1, R_3, R_4, R_9$
• symmetric (8): $R_1, R_2, R_5, R_8, R_9, R_{12}, R_{15}, R_{16}$
• anti-symmetric (12): $R_1-R_8, R_{10}, R_{11}, R_{13}, R_{14}$
• asymmetric (5): $R_1-R_5$
• transitive (13): only $R_9$ and $R_{15}$ are not transitive

Summary:

• The number of distinct binary relations on an n-element set is $2^{(n^2)}$
• The number of reflexive and irreflexive relations is the same.

• The number of strict partial orders (irreflexive transitive relations) is the same as that of partial orders.

• the number of equivalence relations is the number of partitions, which is the Bell number.

Ordered pair $(1,2) = \{\{1\},\{1,2\}\}$

Powerset of A: $P(A)$

$= \{\varnothing, \{1\}, \{2\}, \{1,2\}\}$

$= \{\varnothing, \{1\}, \{2\}, \{1,2\}, \{2,1\}\}$

$= \{\varnothing, \{1\},\{1,2\}, \{2\},\{2,1\}\}$

$= \{\varnothing, (1,2), (2,1) \}$

Ordered pair $(x,y) = \{\{x\},\{x,y\}\}$

if $x=y$ then $(a,a)$

$=\{\{a\},\{a,a\}\}$

$=\{\{a\},\{a\}\}$

$=\{\{a\}\}$

Sets

• A set is an unordered collection of distinct objects (that share some common property) considered as an object in its own right.

• defined intensionaly (semantically): "A set of odd positive ints"

• defined extensionaly (enumeration): $\{a,b,c\}$
• denoted with a capital letter: $A=\{a,b,c\}$, elements with lowercase
• If $a$ is an element of set $B$ then: $a\in B$, if not $a\notin B$
• Order or repetition unimportant: $\{\{a\},b,\varnothing\}=\{b, \{a\},\{\},b,\varnothing,\{\}\}$
• A set can contain other sets, $x \in A, A \in \mathscr{M}$, so $x\in A$ but $x \notin \mathscr{M}$.
• Two sets are equal if they contain the same elements.

Types

• An empty set contains no elements: $\varnothing = \{\}$
• A set, $\{a,b\}$, is distinguished from the ordered pair, $(a,b)$
• $n$-element set is different from ordered $n$-tuple: $\{x_1, x_2,\dots,x_n\} \neq (x_1,\dots, x_n)$
• Two ordered pairs are equal iff: $(a,b) = (x,y) \iff a=x \land b = y$
• The order of the elements or their repetition is of no consequence to sets, $\{a,b,\{\}\}=\{b,\{\},b,\varnothing,a\}$.
• $\{\varnothing\}=\{\{\}\}$

Subset and powerset

• if all elements of a set $A$ are also elements of a set $B$, then $A$ is a subset of $B$: $A \subseteq B$ and $B$ is a superset of $A$: $B \supseteq A$.
• a set $B$ is equal to $A$ if the two sets consist of exactly the same elements: $A = B \iff A \subseteq B \land B \subseteq A$.
• if $A \subseteq B$ but $B \neq A$ then $A$ is a proper subset of $B$: $A \subset B$.
• if a set $P$ contains all possible subsets of a set $A$, then $P$ is a powerset of $A$, denoted as $P(A)$.
• e.g. if $A=\{a,b\}$, then $P=\{\{a\},\{b\},\{ab\},\{\varnothing\}\}$
• here, the cardinality (number of elements) of $P$ is 4, denoted as $|P|=4$

Properties of sets: $\forall A,B,C$

• $A \subseteq A$ (reflexivity)
• $A \subseteq B \land B \subseteq C \rightarrow A \subseteq C$ (transitivity)
• $A \subseteq B \land B \subseteq A \rightarrow A=B$ (anti-symmetry, axiom of extensionality)
• $\varnothing \subseteq A$
• $A\subseteq \varnothing \rightarrow A=\varnothing$