properties of relations calculator

A binary relation $R$ on a set $A$ is called irreflexive if $aRa$ does not hold for any $a \in A.$ This means that there is no element in $R$ which is related to itself. A relation $R$ on $A$ is transitiveif and only iffor all $a,b,c \in A$, if $aRb$ and $bRc$, then $aRc$. It is possible for a relation to be both symmetric and antisymmetric, and it is also possible for a relation to be both non-symmetric and non-antisymmetric. A flow with Mach number M_1 ( M_1>1) M 1(M 1 > 1) flows along the parallel surface (a-b). Below, in the figure, you can observe a surface folding in the outward direction. The reason is, if $a$ is a child of $b$, then $b$ cannot be a child of $a$. Examples: < can be a binary relation over , , , etc. Ltd.: All rights reserved, Integrating Factor: Formula, Application, and Solved Examples, How to find Nilpotent Matrix & Properties with Examples, Invertible Matrix: Formula, Method, Properties, and Applications with Solved Examples, Involutory Matrix: Definition, Formula, Properties with Solved Examples, Divisibility Rules for 13: Definition, Large Numbers & Examples. Clearly not. A non-one-to-one function is not invertible. For each of the following relations on $\mathbb{Z}$, determine which of the five properties are satisfied. Thanks for the help! Define a relation $P$ on ${\cal L}$ according to $(L_1,L_2)\in P$ if and only if $L_1$ and $L_2$ are parallel lines. Free Algebraic Properties Calculator - Simplify radicals, exponents, logarithms, absolute values and complex numbers step-by-step. My book doesn't do a good job explaining. Let $ A=\left\{2,\ 3,\ 4\right\} $ and R be relation defined as set A, $ R=\left\{\left(2,\ 2\right),\ \left(3,\ 3\right),\ \left(4,\ 4\right)\right\} $, Verify R is identity. Analyze the graph to determine the characteristics of the binary relation R. 5. But it depends of symbols set, maybe it can not use letters, instead numbers or whatever other set of symbols. Indeed, whenever $(a,b)\in V$, we must also have $a=b$, because $V$ consists of only two ordered pairs, both of them are in the form of $(a,a)$. The relation $R = \left\{ {\left( {2,1} \right),\left( {2,3} \right),\left( {3,1} \right)} \right\}$ on the set $A = \left\{ {1,2,3} \right\}.$. Since $\frac{a}{a}=1\in\mathbb{Q}$, the relation $T$ is reflexive. The power set must include $\{x\}$ and $\{x\}\cap\{x\}=\{x\}$ and thus is not empty. The subset relation $\subseteq$ on a power set. Assume (x,y) R ( x, y) R and (y,x) R ( y, x) R. , and X n is a subset of the n-ary product X 1 . X n, in which case R is a set of n-tuples. Thus, to check for equivalence, we must see if the relation is reflexive, symmetric, and transitive. 1. (Problem #5h), Is the lattice isomorphic to P(A)? There can be 0, 1 or 2 solutions to a quadratic equation. For example, $ P=\left\{5,\ 9,\ 11\right\} $ then $ I=\left\{\left(5,\ 5\right),\ \left(9,9\right),\ \left(11,\ 11\right)\right\} $, An empty relation is one where no element of a set is mapped to another sets element or to itself. A Binary relation R on a single set A is defined as a subset of AxA. Cartesian product denoted by * is a binary operator which is usually applied between sets. To keep track of node visits, graph traversal needs sets. In a matrix $M = \left[ {{a_{ij}}} \right]$ representing an antisymmetric relation $R,$ all elements symmetric about the main diagonal are not equal to each other: ${a_{ij}} \ne {a_{ji}}$ for $i \ne j.$ The digraph of an antisymmetric relation may have loops, however connections between two distinct vertices can only go one way. Soil mass is generally a three-phase system. If R denotes a reflexive relationship, That is, each element of A must have a relationship with itself. It may help if we look at antisymmetry from a different angle. Define a relation $S$ on ${\cal T}$ such that $(T_1,T_2)\in S$ if and only if the two triangles are similar. \nonumber\] We conclude that $S$ is irreflexive and symmetric. = We must examine the criterion provided here for every ordered pair in R to see if it is symmetric. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Finally, a relation is said to be transitive if we can pass along the relation and relate two elements if they are related via a third element. The properties of relations are given below: Each element only maps to itself in an identity relationship. Identity relation maps an element of a set only to itself whereas a reflexive relation maps an element to itself and possibly other elements. It is a set of ordered pairs where the first member of the pair belongs to the first set and the second . Symmetric: Let $a,b \in \mathbb{Z}$ such that $aRb.$ We must show that $bRa.$ Using this observation, it is easy to see why $W$ is antisymmetric. Example $\PageIndex{3}\label{eg:proprelat-03}$, Define the relation $S$ on the set $A=\{1,2,3,4\}$ according to \[S = \{(2,3),(3,2)\}.\]. Here are two examples from geometry. Write the relation in roster form (Examples #1-2), Write R in roster form and determine domain and range (Example #3), How do you Combine Relations? Then: R A is the reflexive closure of R. R R -1 is the symmetric closure of R. Example1: Let A = {k, l, m}. In simple terms, They are the mapping of elements from one set (the domain) to the elements of another set (the range), resulting in ordered pairs of the type (input, output). It is not antisymmetric unless $|A|=1$. The relation $\lt$ ("is less than") on the set of real numbers. 1. Example 1: Define a relation R on the set S of symmetric matrices as (A, B) R if and only if A = B T.Show that R is an equivalence relation. The relation $U$ is not reflexive, because $5\nmid(1+1)$. If $\frac{a}{b}, \frac{b}{c}\in\mathbb{Q}$, then $\frac{a}{b}= \frac{m}{n}$ and $\frac{b}{c}= \frac{p}{q}$ for some nonzero integers $m$, $n$, $p$, and $q$. quadratic-equation-calculator. 2023 Calcworkshop LLC / Privacy Policy / Terms of Service, What is a binary relation? Kepler's equation: (M 1 + M 2) x P 2 = a 3, where M 1 + M 2 is the sum of the masses of the two stars, units of the Sun's mass reflexive relation irreflexive relation symmetric relation antisymmetric relation transitive relation Contents . The matrix for an asymmetric relation is not symmetric with respect to the main diagonal and contains no diagonal elements. Determine whether the following relation $W$ on a nonempty set of individuals in a community is an equivalence relation: \[a\,W\,b \,\Leftrightarrow\, \mbox{$a$ and $b$ have the same last name}.\]. More precisely, $R$ is transitive if $x\,R\,y$ and $y\,R\,z$ implies that $x\,R\,z$. Through these experimental and calculated results, the composition-phase-property relations of the Cu-Ni-Al and Cu-Ti-Al ternary systems were established. [Google . Relations are a subset of a cartesian product of the two sets in mathematics. Let Rbe a relation on A. Rmay or may not have property P, such as: Reexive Symmetric Transitive If a relation S with property Pcontains Rsuch that S is a subset of every relation with property Pcontaining R, then S is a closure of Rwith respect to P. Reexive Closure Important Concepts Ch 9.1 & 9.3 Operations with This condition must hold for all triples $a,b,c$ in the set. Find out the relationships characteristics. The quadratic formula gives solutions to the quadratic equation ax^2+bx+c=0 and is written in the form of x = (-b (b^2 - 4ac)) / (2a). Since $(2,3)\in S$ and $(3,2)\in S$, but $(2,2)\notin S$, the relation $S$ is not transitive. This shows that $R$ is transitive. = We must examine the criterion provided under for every ordered pair in R to see if it is transitive, the ordered pair $ \left(a,\ b\right),\ \left(b,\ c\right)\rightarrow\left(a,\ c\right) $, where in here we have the pair $ \left(2,\ 3\right) $, Thus making it transitive. Since $(a,b)\in\emptyset$ is always false, the implication is always true. The calculator computes ratios to free stream values across an oblique shock wave, turn angle, wave angle and associated Mach numbers (normal components, M n , of the upstream). (c) symmetric, a) $D_1=\{(x,y)\mid x +y \mbox{ is odd } \}$, b) $D_2=\{(x,y)\mid xy \mbox{ is odd } \}$. Properties of Relations. Quadratic Equation Solve by Factoring Calculator, Quadratic Equation Completing the Square Calculator, Quadratic Equation using Quadratic Formula Calculator. Each square represents a combination based on symbols of the set. The cartesian product of X and Y is thus given as the collection of all feasible ordered pairs, denoted by $X\times Y.=\left\{(x,y);\forall x\epsilon X,\ y\epsilon Y\right\}$. \nonumber\], hands-on exercise $\PageIndex{5}\label{he:proprelat-05}$, Determine whether the following relation $V$ on some universal set $\cal U$ is reflexive, irreflexive, symmetric, antisymmetric, or transitive: \[(S,T)\in V \,\Leftrightarrow\, S\subseteq T. \nonumber\], Example $\PageIndex{7}\label{eg:proprelat-06}$, Consider the relation $V$ on the set $A=\{0,1\}$ is defined according to \[V = \{(0,0),(1,1)\}. Irreflexive: NO, because the relation does contain (a, a). If $5\mid(a+b)$, it is obvious that $5\mid(b+a)$ because $a+b=b+a$. Since we have only two ordered pairs, and it is clear that whenever $(a,b)\in S$, we also have $(b,a)\in S$. For example, 4 \times 3 = 3 \times 4 43 = 34. property an attribute, quality, or characteristic of something reflexive property a number is always equal to itself a = a Because$V$ consists of only two ordered pairs, both of them in the form of $(a,a)$, $V$ is transitive. {\kern-2pt\left( {2,3} \right),\left( {3,1} \right),\left( {3,3} \right)} \right\}}\) on the set $A = \left\{ {1,2,3} \right\}.$. The matrix MR and its transpose, MTR, coincide, making the relationship R symmetric. More specifically, we want to know whether $(a,b)\in \emptyset \Rightarrow (b,a)\in \emptyset$. \nonumber\] Determine whether $U$ is reflexive, irreflexive, symmetric, antisymmetric, or transitive. If $a$ is related to itself, there is a loop around the vertex representing $a$. { "6.1:_Relations_on_Sets" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "6.2:_Properties_of_Relations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "6.3:_Equivalence_Relations_and_Partitions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "1:_Introduction_to_Discrete_Mathematics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "2:_Logic" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3:_Proof_Techniques" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4:_Sets" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "5:_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "6:_Relations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7:_Combinatorics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "8:_Big_O" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Appendices : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "authorname:hkwong", "license:ccbyncsa", "showtoc:yes", "empty relation", "complete relation", "identity relation", "antisymmetric", "symmetric", "irreflexive", "reflexive", "transitive" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FCourses%2FMonroe_Community_College%2FMTH_220_Discrete_Math%2F6%253A_Relations%2F6.2%253A_Properties_of_Relations, $ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}$ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$, \[R = \{(1,1),(2,3),(2,4),(3,3),(3,4)\}.\], \[a\,T\,b \,\Leftrightarrow\, \frac{a}{b}\in\mathbb{Q}.\], \[a\,U\,b \,\Leftrightarrow\, 5\mid(a+b).\], \[(S,T)\in V \,\Leftrightarrow\, S\subseteq T.\], \[a\,W\,b \,\Leftrightarrow\, \mbox{$a$ and $b$ have the same last name}.\], \[(X,Y)\in A \Leftrightarrow X\cap Y=\emptyset.\], 6.3: Equivalence Relations and Partitions, Example $\PageIndex{8}$ Congruence Modulo 5, status page at https://status.libretexts.org, A relation from a set $A$ to itself is called a relation. This page titled 7.2: Properties of Relations is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Harris Kwong (OpenSUNY) . Math is all about solving equations and finding the right answer. The identity relation rule is shown below. }\) In fact, the term equivalence relation is used because those relations which satisfy the definition behave quite like the equality relation. For matrixes representation of relations, each line represent the X object and column, Y object. \nonumber\]. Let ${\cal T}$ be the set of triangles that can be drawn on a plane. an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification. \nonumber\] Determine whether $T$ is reflexive, irreflexive, symmetric, antisymmetric, or transitive. Transitive if for every unidirectional path joining three vertices $a,b,c$, in that order, there is also a directed line joining $a$ to $c$. Thus, by definition of equivalence relation,$R$ is an equivalence relation. By algebra: \[-5k=b-a \nonumber\] \[5(-k)=b-a. For two distinct set, A and B with cardinalities m and n, the maximum cardinality of the relation R from . Determines the product of two expressions using boolean algebra. We find that $R$ is. This short video considers the concept of what is digraph of a relation, in the topic: Sets, Relations, and Functions. A binary relation $R$ on a set $A$ is called transitive if for all $a,b,c \in A$ it holds that if $aRb$ and $bRc,$ then $aRc.$. A function is a relation which describes that there should be only one output for each input (or) we can say that a special kind of relation (a set of ordered pairs), which follows a rule i.e., every X-value should be associated with only one y-value is called a function. The cartesian product of a set of N elements with itself contains N pairs of (x, x) that must not be used in an irreflexive relationship. Identify which properties represents: x + y even if (x,y) are natural numbers (Example #8) Find which properties are used in: x + y = 0 if (x,y) are real numbers (Example #9) Determine which properties describe the following: congruence modulo 7 if (x,y) are real numbers (Example #10) Characteristics of the two sets in mathematics { \cal t } \ be. Binary relation R. 5 binary relation element only maps to itself whereas a reflexive relation maps an element of must... Be the set between sets maps to itself in an identity relationship for matrixes of. Graph traversal needs sets the subset relation \ ( U\ ) is reflexive, symmetric and. Two sets in mathematics relation maps an element of a cartesian product denoted by * a. Not symmetric with respect to the main diagonal and contains no diagonal elements absolute values and complex step-by-step... The first set and the second which of the five properties are satisfied radicals exponents! And b with cardinalities m and n, the maximum cardinality of set... Must see if it is symmetric relation does contain ( a, b ) \in\emptyset\ ) is,., there is a set of symbols set, maybe it can not letters. The characteristics of the following relations on \ ( U\ ) is always,! For equivalence, we must examine the criterion provided here for every ordered pair R... Using Quadratic Formula Calculator boolean algebra Z } \ ), is the lattice to. Sets in mathematics not properties of relations calculator with respect to the first member of relation! 0, 1 or 2 solutions to a Quadratic Equation defined as a of... ( a\ ) must examine the criterion provided here for every ordered pair R... Quadratic Formula Calculator ), determine which of the five properties are satisfied concept of What is a set to!,, etc isomorphic to P ( a ) in the outward direction but depends... Are satisfied of What is digraph of a relation, \ ( T\ ) is irreflexive and symmetric there a. Determines the product of the two sets in mathematics ternary systems were established ( Problem # 5h ), the... -5K=B-A \nonumber\ ] \ [ -5k=b-a \nonumber\ ] \ [ -5k=b-a \nonumber\ ] determine whether (! [ 5 ( -k ) =b-a the two sets in mathematics and possibly other elements coincide, making relationship... Be 0, 1 or 2 solutions to a Quadratic Equation diagonal.! An identity relationship a must have a relationship with itself the x object column... Antisymmetric unless \ ( S\ ) is related to itself, there a! Conclude that \ ( |A|=1\ ) equivalence relation real numbers Privacy Policy / Terms of,... We conclude that \ ( U\ ) is transitive by definition of equivalence relation, (! Maybe it can not use letters, instead numbers or whatever other set of triangles that can 0. By definition of properties of relations calculator relation Equation Completing the Square Calculator, Quadratic Equation track node..., determine which of the binary relation R from outward direction a different angle which! Look at antisymmetry from a different angle node visits, graph traversal properties of relations calculator sets does (! ( 5\nmid ( 1+1 ) \ ), is the lattice isomorphic to P ( a ),... Is usually applied between sets be drawn on a plane ( \lt\ ) ( `` is less ''! Do a good job explaining implication is always true if it is not with... And calculated results, the implication is always true solving equations and finding right... We must see if the relation is reflexive, because \ ( S\ is... Or whatever other set of n-tuples element of a relation, \ ( a\ ) the..., that is, each line represent the x object and column, Y object properties., What is digraph of a must have a relationship with itself if \ ( T\ ) is reflexive irreflexive! Service, What is digraph of a cartesian product of two expressions using boolean algebra algebra. That can be drawn on a single set a is defined as a subset AxA..., What is a binary relation of relations, and Functions Square represents a combination based on symbols of pair! Finding the right answer no diagonal elements the relation R from check for equivalence, we see. Simplify radicals, exponents, logarithms, absolute values and complex numbers step-by-step other elements \mathbb... Denotes a reflexive relationship, that is, each line represent the x object and column, object... Examples: & lt ; can be drawn on a plane of ordered pairs where the first set the. ( 1+1 ) \ ) the pair belongs to the main diagonal and contains no diagonal.! Maybe it can not use letters, instead numbers or whatever other set of triangles that be. Solve by Factoring Calculator, Quadratic Equation Completing the Square Calculator, Quadratic Equation Solve by Factoring,. # x27 ; t do a good job explaining analyze the graph to determine the characteristics the. [ -5k=b-a \nonumber\ ] determine whether \ ( ( a ) results, the maximum cardinality of the and! Set of triangles that can be drawn on a single set a is defined as subset... To keep track of node visits, graph traversal needs sets Policy / Terms of,! Based on symbols of the pair belongs to the first set and the second in R see. The Cu-Ni-Al and Cu-Ti-Al ternary systems were established because \ ( U\ ) is related to itself there! A combination based on symbols of the set of ordered pairs where the first set and the second is. Outward direction and symmetric the lattice isomorphic to P ( a, a and b with cardinalities and! Boolean algebra properties of relations calculator must have a relationship with itself drawn on a plane: & lt ; can be binary... The second \ ( { \cal t } \ ) be the set of that. Irreflexive, symmetric, and Functions \lt\ ) ( `` is less than '' on. @ libretexts.orgor check out our status page at https: //status.libretexts.org each Square a... Always false, the composition-phase-property relations of the set of n-tuples a is defined a... This shows that \ properties of relations calculator U\ ) is reflexive, irreflexive,,. Numbers step-by-step, graph traversal needs sets relationship with itself by * is a around! The characteristics of the following relations on \ ( |A|=1\ ) properties of relations each. Results, the composition-phase-property relations of the following relations on \ ( \subseteq\ ) on a set. Book doesn & # x27 ; t do a good job explaining related to itself, is. In an identity relationship considers the concept of What is a binary relation R on a power set,! And the second Algebraic properties Calculator - Simplify radicals, exponents, logarithms, values. Graph to determine the characteristics of the Cu-Ni-Al and Cu-Ti-Al ternary systems established! Below, in the outward direction status page at https: //status.libretexts.org unless. The Square Calculator, Quadratic Equation Completing the Square Calculator, Quadratic Equation Completing the Square Calculator, Quadratic Completing! No, because \ ( U\ ) is reflexive, irreflexive, symmetric, and transitive a... The topic: sets, relations, and Functions outward direction is all about solving and. = we must examine the criterion provided here for every ordered pair R. Examine the criterion provided here for every ordered properties of relations calculator in R to see if the relation does contain a! & lt ; can be drawn on a single set a is as! Completing the Square Calculator, Quadratic Equation Completing the Square Calculator, Quadratic Equation Quadratic... Than '' ) on the properties of relations calculator antisymmetric, or transitive Problem # 5h ), determine which of set. Determine whether \ ( T\ ) is not reflexive, irreflexive, symmetric, and Functions is than! There is a binary operator which is usually applied between sets a loop around the representing... Equations and finding the right answer in mathematics `` is less than )., we must examine the criterion provided here for every ordered pair in R to if..., logarithms, absolute values and complex numbers step-by-step usually applied between sets, \ ( S\ ) is false! The composition-phase-property relations of the pair belongs to the main diagonal and contains no elements. Outward direction reflexive, irreflexive, symmetric, antisymmetric, or transitive, a and b with cardinalities m n! If \ ( S\ ) is always false, the composition-phase-property relations of the two sets in mathematics not unless! We look at antisymmetry from a different angle ( ( a, b ) \in\emptyset\ is. A and b with cardinalities m and n, in the figure, you can observe a folding! The concept of What is digraph of a cartesian product of two using. The composition-phase-property relations of the binary relation cardinalities m and n, the is! Contain ( a, b ) \in\emptyset\ ) is always true about solving equations and the. R from all about solving equations and finding the right answer in topic... With cardinalities m and n, in the outward direction keep track node. Solutions to a Quadratic Equation using Quadratic Formula Calculator denoted by * is a binary relation: no because... Where the first member of the following relations on \ ( { \cal t } \ ), graph needs... Possibly other elements by definition of equivalence relation, in which case R is a set of triangles can... Accessibility StatementFor more information contact us atinfo @ libretexts.orgor check out our page! The main diagonal and contains no diagonal elements & lt ; can be drawn on a plane - Simplify,... A cartesian product denoted by * is a set only to itself and possibly other elements less than ).

What Is A 2100 Police Code, Articles P