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--- sav08:sets_and_relations [2010/02/26 22:03]
vkuncak
+++ sav08:sets_and_relations [2015/04/21 17:30]
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-====== Sets and Relations ======
-===== Sets =====
-Sets are unordered collection of elements.
-We denote a finite set containing only elements $a$, $b$ and $c$ by $\{ a, b, c \}$.  The order and number of occurrences does not matter:
-$\{ a, b, c \} = \{ c, a, b \} = \{ a, b, b, c \}$.
-  * $a \in \{ a,b,c \}$
-  * $d \notin \{a,b,c\}$ iff $d \neq a \land d \neq b \land d \neq c$
-Empty set: $\emptyset$. For every $x$ we have $x \notin \emptyset$.
-To denote large or infinite sets we can use set comprehensions: $\{ x.\ P(x) \}$ is set of all objects with property $P$.
-\[
-    y \in \{ x. P(x) \} \ \leftrightarrow\ P(y)
-\]
-Notation for set comprehension: $\{ f(x)|x. P(x) \} = \{ y. (\exists x. y=f(x) \land P(x)) \}$
-Sometimes the binder $x$ can be inferred from context so we write simply $\{ f(x) | P(x) \}$.  In general there is ambiguity in which variables are bound. (Example: what does the $a$ in $f(a,b)$ refer to in the expression:
-\[
-   \{a \} \cup \{ f(a,b) \mid P(a,b) \}
-\]
-does it refer to the outerone $a$ as in $\{a\}$ or is it a newly bound variable? The notation with dot and bar resolves this ambiguity.
-Subset: $A \subseteq B$ means $\forall x. x \in A \rightarrow x \in B$
-\[
-   A \cup B = \{ x. x \in A \lor x \in B \}
-\]
-\[
-   A \cap B = \{ x. x \in A \land x \in B \}
-\]
-\[
-   A \setminus B = \{ x. x \in A \land x \notin B \}
-\]
-Boolean algebra of subsets of some set $U$ (we define $A^c = U \setminus A$):
-  * $\cup, \cap$ are associative, commutative, idempotent
-  * neutral and zero elements: $A \cup \emptyset = A$, $A \cap \emptyset = \emptyset$
-  * absorption: $A \cup A = A$, $A \cap A = A$
-  * deMorgan laws: $(A \cup B)^c = A^c \cap B^c$, $(A \cap B)^c = A^c \cup B^c$
-  * complement as partition of universal set: $A \cap A^c = \emptyset$, $A \cup A^c = U$
-  * double complement: $(A^c)^c = A$
-Which axioms are sufficient?
-  * [[http://citeseer.ist.psu.edu/57459.html|William McCune: Solution of the Robbins Problem. J. Autom. Reasoning 19(3): 263-276 (1997)]]
-==== Infinte Unions and Intersections ====
-Note that sets can be nested. Consider, for example, the following set $S$
-\[
-   S = \{ \{ p, \{q, r\} \}, r \}
-\]
-This set has two elements. The first element is another set. We have $\{ p, \{q, r\} \} \in S$. Note that it is not the case that
-Suppose that we have a set $B$ that contains other sets.  We define union of the sets contained in $B$ as follows:
-\[
-   \bigcup B = \{ x.\ \exists a. a \in B \land x \in a \}
-\]
-As a special case, we have
-\[
-   \bigcup \{ a_1, a_2, a_3 \} = a_1 \cup a_2 \cup a_3
-\]
-Often the elements of the set $B$ are computed by a set comprehension of the form $B = \{ f(i).\ i \in J \}$.
-We then write
-\[
-   \bigcup_{i \in J} f(i)
-\]
-and the meaning is
-\[
-   \bigcup \{ f(i).\ i \in J \}
-\]
-Therefore, $x \in \bigcup \{ f(i).\ i \in J \}$ is equivalent to $\exists i.\ i \in J \land x \in f(i)$.
-We analogously define intersection of elements in the set:
-\[
-   \bigcap B = \{ x. \forall a. a \in B \rightarrow x \in a \}
-\]
-As a special case, we have
-\[
-   \bigcap \{ a_1, a_2, a_3 \} = a_1 \cap a_2 \cap a_3
-\]
-We similarly define intersection of an infinite family
-\[
-   \bigcap_{i \in J} f(i)
-\]
-and the meaning is
-\[
-   \bigcap \{ f(i).\ i \in J \}
-\]
-Therefore, $x \in \bigcap \{ f(i).\ i \in J \}$ is equivalent to $\forall i.\ i \in J \rightarrow x \in f(i)$.
-===== Relations =====
-Pairs:
-\[
-    (a,b) = (u,v)  \iff (a = u \land b = v)
-\]
-Cartesian product:
-\[
-    A \times B = \{ (x,y) \mid x \in A \land y \in B \}
-\]
-Relations $r$ is simply a subset of $A \times B$, that is $r \subseteq A \times B$.
-Note:
-\[
-   A \times (B \cap C) = (A \times B) \cap (A \times C)
-\]
-\[
-   A \times (B \cup C) = (A \times B) \cup (A \times C)
-\]
-=== Diagonal relation ===
-$\Delta_A \subseteq A \times A$, is given by
-\[
-   \Delta_A = \{(x,x) \mid x \in A\}
-\]
-==== Set operations ====
-Relations are sets of pairs, so operations $\cap, \cup, \setminus$ apply.
-==== Relation Inverse ====
-\[
-    r^{-1} = \{(y,x) \mid (x,y) \in r \}
-\]
-==== Relation Composition ====
-\[
-   r_1 \circ r_2 = \{ (x,z) \mid \exists y. (x,y) \in r_1 \land (y,z) \in r_2\}
-\]
-Note: relations on a set $A$ together with relation composition and $\Delta_A$ form a //monoid// structure:
-\[
-\begin{array}{l}
-   r_1 \circ (r_2 \circ r_3) = (r_1 \circ r_2) \circ r_3 \\
-   r \circ \Delta_A = r = \Delta_A \circ r
-\end{array}
-\]
-Moreover,
-\[
-   \emptyset \circ r = \emptyset = r \circ \emptyset
-\]
-\[
-    r_1 \subseteq r_2 \rightarrow r_1 \circ s \subseteq r_2 \circ s
-\]
-\[
-    r_1 \subseteq r_2 \rightarrow s \circ r_1 \subseteq s \circ r_2
-\]
-==== Relation Image ====
-When $S \subseteq A$ and $r \subseteq A \times A$ we define **image** of a set $S$ under relation $A$ as
-\[
-   S\bullet r = \{ y.\ \exists x. x \in S \land (x,y) \in r \}
-\]
-==== Transitive Closure ====
-Iterated composition let $r \subseteq A \times A$.
-\[
-\begin{array}{l}
-  r^0 = \Delta_A \\
-  r^{n+1} = r \circ r^n
-\end{array}
-\]
-So, $r^n$ is n-fold composition of relation with itself.
-Transitive closure:
-\[
-   r^* = \bigcup_{n \geq 0} r^n
-\]
-Equivalent statement: $r^*$ is equal to the least relation $s$ (with respect to $\subseteq$) that satisfies
-\[
-    \Delta_A\ \cup\ (s \circ r)\ \subseteq\ s
-\]
-or, equivalently, the least relation $s$ (with respect to $\subseteq$) that satisfies
-\[
-    \Delta_A\ \cup\ (r \circ s)\ \subseteq\ s
-\]
-or, equivalently, the least relation $s$ (with respect to $\subseteq$) that satisfies
-\[
-    \Delta_A\ \cup\ r \cup (s \circ s)\ \subseteq\ s
-\]
-==== Some Laws in Algebra of Relations ====
-\[
-    (r_1 \circ r_2)^{-1} = r_2^{-1} \circ r_1^{-1}
-\]
-\[
-    r_1 \circ (r_2 \cup r_3) = (r_1 \circ r_2) \cup (r_1 \circ r_3)
-\]
-\[
-    (r^{-1})^{*} = (r^{*})^{-1}
-\]
-Binary relation $r \subseteq A\times A$ can be represented as a directed graph $(A,r)$ with nodes $A$ and edges $r$
-  * Graphical representation of $r^{-1}$, $r^{*}$, and $(r \cup r^{-1})^{*}$
-Equivalence relation $r$ is relation with these properties:
-  * reflexive: $\Delta_A \subseteq r$
-  * symmetric: $r^{-1} \subseteq r$
-  * transitive: $r \circ r \subseteq r$
-Equivalence classes are defined by
-\[
-   x/r = \{y \mid (x,y) \in r
-\]
-The set $\{ x/r \mid x \in A \}$ is a partition:
-  * each set non-empty
-  * sets are disjoint
-  * their union is $A$
-Conversely: each collection of sets $P$ that is a partition defines equivalence class by
-\[
-   r = \{ (x,y) \mid \exists c \in P. x \in c \land y \in c \}
-\]
-Congruence: equivalence that agrees with some set of operations.
-Partial orders:
-  * reflexive
-  * antisymmetric: $r \cap r^{-1} \subseteq \Delta_A$
-  * transitive
-===== Functions =====
-**Example:** an example function $f : A \to B$ for $A = \{a,b,c\}$, $B=\{1,2,3\}$ is
-\[
-  f = \{ (a,3), (b,2), (c,3) \}
-\]
-Definition of function, injectivity, surjectivity.
-$2^B = \{ A \mid A \subseteq B \}$
-$(A \to B) = B^A$ - the set of all functions from $A$ to $B$.  For $|B|>2$ it is a strictly bigger set than $B$.
-$(A \to B \to C) = (A \to (B \to C))$ (think of exponentiation on numbers)
-Note that $A \to B \to C$ is [[isomorphism|isomorphic]] to $A \times B \to C$, they are two ways of representing functions with two arguments.  $(C^B)^A = C^{B \times A}$
-There is also isomorphism between
-  * n-tuples $(x_1,\ldots,x_n) \in A^n$ and
-  * functions $f : \{1,\ldots,n\} \to A$, where $f = \{(1,x_1),\ldots,(n,x_n) \}$
-==== Function update =====
-Function update operator takes a function $f : A \to B$ and two values $a_0 \in A$, $b_0 \in B$ and creates a new function $f[a_0 \mapsto b_0]$ that behaves like $f$ in all points except at $a_0$, where it has value $b_0$.  Formally,
-\[
-f[a_0 \mapsto b_0](x) = \left\{\begin{array}{l}
-  b_0, \mbox{ if } x=a_0 \\
-  f(x), \mbox{ if } x \neq a_0
-\]
-==== Domain and Range of Relations and Functions ====
-For relation $r \subseteq A \times B$ we define domain and range of $r$:
-\[
-    dom(r) = \{ x.\ \exists y. (x,y) \in r \}
-\]
-\[
-    ran(r) = \{ y.\ \exists x. (x,y) \in r \}
-\]
-Clearly, $dom(r) \subseteq A$ and $ran(r) \subseteq B$.
-==== Partial Function ====
-Notation: $\exists^{\leq 1} x. P(x)$ means $\forall x. \forall y. (P(x) \land P(y))\rightarrow x=y$.
-**Partial function** $f : A \hookrightarrow B$ is relation $f \subseteq A \times B$ such that
-\[
-    \forall x \in A. \exists^{\le 1} y.\ (x,y)\in f
-\]
-Generalization of function update is override of partial functions, $f \oplus g$
-==== Range, Image, and Composition ====
-The following properties follow from the definitions:
-\[
-   (S \bullet r_1) \bullet r_2 = S \bullet (r_1 \circ r_2)
-\]
-\[
-   S \bullet r = ran(\Delta_S \circ r)
-\]
-===== Further references =====
-  * [[sav08:discrete_mathematics_by_rosen|Discrete Mathematics by Rosen]]
-  * [[:Gallier Logic Book]], Chapter 2