Here are two examples from geometry. + For each of the following relations on \(\mathbb{N}\), determine which of the three properties are satisfied. Then there are and so that and . Suppose divides and divides . Since\(aRb\),\(5 \mid (a-b)\) by definition of \(R.\) Bydefinition of divides, there exists an integer \(k\) such that \[5k=a-b. It is easy to check that \(S\) is reflexive, symmetric, and transitive. \nonumber\] Determine whether \(S\) is reflexive, irreflexive, symmetric, antisymmetric, or transitive. hands-on exercise \(\PageIndex{4}\label{he:proprelat-04}\). ) R , then (a 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 \(q\). <> [2], Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. "is ancestor of" is transitive, while "is parent of" is not. Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. Example \(\PageIndex{4}\label{eg:geomrelat}\). For example, the relation "is less than" on the natural numbers is an infinite set Rless of pairs of natural numbers that contains both (1,3) and (3,4), but neither (3,1) nor (4,4). Get more out of your subscription* Access to over 100 million course-specific study resources; 24/7 help from Expert Tutors on 140+ subjects; Full access to over 1 million Textbook Solutions Transitive if \((M^2)_{ij} > 0\) implies \(m_{ij}>0\) whenever \(i\neq j\). Then \(\frac{a}{c} = \frac{a}{b}\cdot\frac{b}{c} = \frac{mp}{nq} \in\mathbb{Q}\). For the relation in Problem 8 in Exercises 1.1, determine which of the five properties are satisfied. Has 90% of ice around Antarctica disappeared in less than a decade? . \(-k \in \mathbb{Z}\) since the set of integers is closed under multiplication. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Note that 2 divides 4 but 4 does not divide 2. It is clearly symmetric, because \((a,b)\in V\) always implies \((b,a)\in V\). \nonumber\] It is clear that \(A\) is symmetric. For the relation in Problem 9 in Exercises 1.1, determine which of the five properties are satisfied. However, \(U\) is not reflexive, because \(5\nmid(1+1)\). For most common relations in mathematics, special symbols are introduced, like "<" for "is less than", and "|" for "is a nontrivial divisor of", and, most popular "=" for "is equal to". The complete relation is the entire set \(A\times A\). z Quasi-reflexive: If each element that is related to some element is also related to itself, such that relation ~ on a set A is stated formally: a, b A: a ~ b (a ~ a b ~ b). Note2: r is not transitive since a r b, b r c then it is not true that a r c. Since no line is to itself, we can have a b, b a but a a. hands-on exercise \(\PageIndex{2}\label{he:proprelat-02}\). CS202 Study Guide: Unit 1: Sets, Set Relations, and Set. \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)\}. Properties of Relations in Discrete Math (Reflexive, Symmetric, Transitive, and Equivalence) Intermation Types of Relations || Reflexive || Irreflexive || Symmetric || Anti Symmetric ||. Transitive: Let \(a,b,c \in \mathbb{Z}\) such that \(aRb\) and \(bRc.\) We must show that \(aRc.\) Which of the above properties does the motherhood relation have? For the relation in Problem 6 in Exercises 1.1, determine which of the five properties are satisfied. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. 12_mathematics_sp01 - Read online for free. q This shows that \(R\) is transitive. The complete relation is the entire set A A. Write the definitions of reflexive, symmetric, and transitive using logical symbols. For the relation in Problem 7 in Exercises 1.1, determine which of the five properties are satisfied. For relation, R, an ordered pair (x,y) can be found where x and y are whole numbers and x is divisible by y. For example, "is less than" is irreflexive, asymmetric, and transitive, but neither reflexive nor symmetric, may be replaced by {\displaystyle R\subseteq S,} The identity relation consists of ordered pairs of the form (a, a), where a A. Symmetric if every pair of vertices is connected by none or exactly two directed lines in opposite directions. To help Teachoo create more content, and view the ad-free version of Teachooo please purchase Teachoo Black subscription. Antisymmetric: For al s,t in B, if sGt and tGs then S=t. s > t and t > s based on definition on B this not true so there s not equal to t. Therefore not antisymmetric?? (14, 14) R R is not reflexive Check symmetric To check whether symmetric or not, If (a, b) R, then (b, a) R Here (1, 3) R , but (3, 1) R R is not symmetric Check transitive To check whether transitive or not, If (a,b) R & (b,c) R , then (a,c) R Here, (1, 3) R and (3, 9) R but (1, 9) R. R is not transitive Hence, R is neither reflexive, nor . that is, right-unique and left-total heterogeneous relations. Exercise \(\PageIndex{12}\label{ex:proprelat-12}\). The first condition sGt is true but tGs is false so i concluded since both conditions are not met then it cant be that s = t. so not antisymmetric, reflexive, symmetric, antisymmetric, transitive, We've added a "Necessary cookies only" option to the cookie consent popup. x Therefore\(U\) is not an equivalence relation, Determine whether the following relation \(V\) on some universal set \(\cal U\) is an equivalence relation: \[(S,T)\in V \,\Leftrightarrow\, S\subseteq T.\], Example \(\PageIndex{7}\label{eg:proprelat-06}\), Consider the relation \(V\) on the set \(A=\{0,1\}\) is defined according to \[V = \{(0,0),(1,1)\}.\]. Then , so divides . It only takes a minute to sign up. 3 David Joyce I'm not sure.. Exercise. R and hands-on exercise \(\PageIndex{6}\label{he:proprelat-06}\), Determine whether the following relation \(W\) on a nonempty set of individuals in a community is reflexive, irreflexive, symmetric, antisymmetric, or transitive: \[a\,W\,b \,\Leftrightarrow\, \mbox{$a$ and $b$ have the same last name}. Example 6.2.5 : How to prove a relation is antisymmetric \(5 \mid (a-b)\) and \(5 \mid (b-c)\) by definition of \(R.\) Bydefinition of divides, there exists an integers \(j,k\) such that \[5j=a-b. Given that \( A=\emptyset \), find \( P(P(P(A))) 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. and By algebra: \[-5k=b-a \nonumber\] \[5(-k)=b-a. \nonumber\]. y This operation also generalizes to heterogeneous relations. Consider the relation \(R\) on \(\mathbb{Z}\) defined by \(xRy\iff5 \mid (x-y)\). ) R & (b A binary relation G is defined on B as follows: for The relation \(V\) is reflexive, because \((0,0)\in V\) and \((1,1)\in V\). 7. Example \(\PageIndex{3}\label{eg:proprelat-03}\), Define the relation \(S\) on the set \(A=\{1,2,3,4\}\) according to \[S = \{(2,3),(3,2)\}.\]. Since we have only two ordered pairs, and it is clear that whenever \((a,b)\in S\), we also have \((b,a)\in S\). 1. Symmetric: Let \(a,b \in \mathbb{Z}\) such that \(aRb.\) We must show that \(bRa.\) Define a relation \(S\) on \({\cal T}\) such that \((T_1,T_2)\in S\) if and only if the two triangles are similar. Then , so divides . Co-reflexive: A relation ~ (similar to) is co-reflexive for all . It is transitive if xRy and yRz always implies xRz. Define the relation \(R\) on the set \(\mathbb{R}\) as \[a\,R\,b \,\Leftrightarrow\, a\leq b.\] Determine whether \(R\) is reflexive, symmetric,or transitive. If a law is new but its interpretation is vague, can the courts directly ask the drafters the intent and official interpretation of their law? y \(\therefore R \) is transitive. Definition. \nonumber\]. Show that `divides' as a relation on is antisymmetric. 1. If x < y, and y < z, then it must be true that x < z. Equivalence Relations The properties of relations are sometimes grouped together and given special names. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We conclude that \(S\) is irreflexive and symmetric. The topological closure of a subset A of a topological space X is the smallest closed subset of X containing A. Set Notation. At what point of what we watch as the MCU movies the branching started? Define a relation P on L according to (L1, L2) P if and only if L1 and L2 are parallel lines. A relation R R in the set A A is given by R = \ { (1, 1), (2, 3), (3, 2), (4, 3), (3, 4) \} R = {(1,1),(2,3),(3,2),(4,3),(3,4)} The relation R R is Choose all answers that apply: Reflexive A Reflexive Symmetric B Symmetric Transitive C 4 0 obj Now we are ready to consider some properties of relations. a function is a relation that is right-unique and left-total (see below). The relation \(R\) is said to be antisymmetric if given any two. and caffeine. A relation on the set A is an equivalence relation provided that is reflexive, symmetric, and transitive. The relation \(U\) on the set \(\mathbb{Z}^*\) is defined as \[a\,U\,b \,\Leftrightarrow\, a\mid b. It is not antisymmetric unless \(|A|=1\). a) \(B_1=\{(x,y)\mid x \mbox{ divides } y\}\), b) \(B_2=\{(x,y)\mid x +y \mbox{ is even} \}\), c) \(B_3=\{(x,y)\mid xy \mbox{ is even} \}\), (a) reflexive, transitive If a relation \(R\) on \(A\) is both symmetric and antisymmetric, its off-diagonal entries are all zeros, so it is a subset of the identity relation. Explain why none of these relations makes sense unless the source and target of are the same set. (b) reflexive, symmetric, transitive Antisymmetric if \(i\neq j\) implies that at least one of \(m_{ij}\) and \(m_{ji}\) is zero, that is, \(m_{ij} m_{ji} = 0\). Give reasons for your answers and state whether or not they form order relations or equivalence relations. As another example, "is sister of" is a relation on the set of all people, it holds e.g. Likewise, it is antisymmetric and transitive. Similarly and = on any set of numbers are transitive. If \(a\) is related to itself, there is a loop around the vertex representing \(a\). Hence, these two properties are mutually exclusive. . Are there conventions to indicate a new item in a list? Eon praline - Der TOP-Favorit unserer Produkttester. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. (b) Consider these possible elements ofthe power set: \(S_1=\{w,x,y\},\qquad S_2=\{a,b\},\qquad S_3=\{w,x\}\). But a relation can be between one set with it too. Or similarly, if R (x, y) and R (y, x), then x = y. Let A be a nonempty set. Example \(\PageIndex{2}\label{eg:proprelat-02}\), Consider the relation \(R\) on the set \(A=\{1,2,3,4\}\) defined by \[R = \{(1,1),(2,3),(2,4),(3,3),(3,4)\}. (b) is neither reflexive nor irreflexive, and it is antisymmetric, symmetric and transitive. Formally, X = { 1, 2, 3, 4, 6, 12 } and Rdiv = { (1,2), (1,3), (1,4), (1,6), (1,12), (2,4), (2,6), (2,12), (3,6), (3,12), (4,12) }. Is there a more recent similar source? The best answers are voted up and rise to the top, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Let $aA$ and $R = f (a)$ Since R is reflexive we know that $\forall aA \,\,\,,\,\, \exists (a,a)R$ then $f (a)= (a,a)$ Let \({\cal L}\) be the set of all the (straight) lines on a plane. We have \((2,3)\in R\) but \((3,2)\notin R\), thus \(R\) is not symmetric. So, congruence modulo is reflexive. (Python), Chapter 1 Class 12 Relation and Functions. 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 \(q\). y x x The relation \(U\) is not reflexive, because \(5\nmid(1+1)\). Is this relation transitive, symmetric, reflexive, antisymmetric? Given sets X and Y, a heterogeneous relation R over X and Y is a subset of { (x,y): xX, yY}. A relation on a finite set may be represented as: For example, on the set of all divisors of 12, define the relation Rdiv by. 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)\}. We'll show reflexivity first. If R is a binary relation on some set A, then R has reflexive, symmetric and transitive closures, each of which is the smallest relation on A, with the indicated property, containing R. Consequently, given any relation R on any . Consider the following relation over is (choose all those that apply) a. Reflexive b. Symmetric c. Transitive d. Antisymmetric e. Irreflexive 2. a) \(A_1=\{(x,y)\mid x \mbox{ and } y \mbox{ are relatively prime}\}\). Let's take an example. whether G is reflexive, symmetric, antisymmetric, transitive, or none of them. Strange behavior of tikz-cd with remember picture. And the symmetric relation is when the domain and range of the two relations are the same. Learn more about Stack Overflow the company, and our products. Relationship between two sets, defined by a set of ordered pairs, This article is about basic notions of relations in mathematics. No, is not symmetric. Projective representations of the Lorentz group can't occur in QFT! Definition: equivalence relation. = methods and materials. Antisymmetric relation is a concept of set theory that builds upon both symmetric and asymmetric relation in discrete math. Hence, \(S\) is symmetric. Transitive: If any one element is related to a second and that second element is related to a third, then the first element is related to the third. = Exercise \(\PageIndex{5}\label{ex:proprelat-05}\). So Congruence Modulo is symmetric. For transitivity the claim should read: If $s>t$ and $t>u$, becasue based on the definition the number of 0s in s is greater than the number of 0s in t.. so isn't it suppose to be the > greater than sign. \(\therefore R \) is symmetric. He has been teaching from the past 13 years. Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors. For each relation in Problem 1 in Exercises 1.1, determine which of the five properties are satisfied. [1] (c) symmetric, a) \(D_1=\{(x,y)\mid x +y \mbox{ is odd } \}\), b) \(D_2=\{(x,y)\mid xy \mbox{ is odd } \}\). Instead, it is irreflexive. Duress at instant speed in response to Counterspell, Dealing with hard questions during a software developer interview, Partner is not responding when their writing is needed in European project application. No, Jamal can be the brother of Elaine, but Elaine is not the brother of Jamal. Exercise \(\PageIndex{3}\label{ex:proprelat-03}\). The functions should behave like this: The input to the function is a relation on a set, entered as a dictionary. How do I fit an e-hub motor axle that is too big? Exercise. The concept of a set in the mathematical sense has wide application in computer science. The notations and techniques of set theory are commonly used when describing and implementing algorithms because the abstractions associated with sets often help to clarify and simplify algorithm design. y It is clearly reflexive, hence not irreflexive. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Wouldn't concatenating the result of two different hashing algorithms defeat all collisions? Reflexive Relation Characteristics. N x Since \((2,2)\notin R\), and \((1,1)\in R\), the relation is neither reflexive nor irreflexive. Varsity Tutors does not have affiliation with universities mentioned on its website. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. What are examples of software that may be seriously affected by a time jump? [vj8&}4Y1gZ] +6F9w?V[;Q wRG}}Soc);q}mL}Pfex&hVv){2ks_2g2,7o?hgF{ek+ nRr]n 3g[Cv_^]+jwkGa]-2-D^s6k)|@n%GXJs P[:Jey^+r@3 4@yt;\gIw4['2Twv%ppmsac =3. Not affiliated with Varsity Tutors the set of ordered pairs, This article is about basic notions of in. To ( L1, L2 ) P if and only if L1 and L2 are lines. Has wide application in computer Science ' as a relation can be the brother Elaine... Two Sets, set relations, and 1413739 range of the five properties are satisfied ) if! ( y, x ), Chapter 1 Class 12 relation and Functions motor axle that is,. ) since the set of integers is closed under multiplication not antisymmetric unless \ ( S\ is... On L according to ( L1, L2 ) P if and only if L1 and L2 parallel. Relation is the smallest closed subset of x containing a or none of them the two relations the. Asymmetric relation in Problem 1 in Exercises 1.1, determine which of five... Notions of relations in mathematics and = on any set of all people, it holds e.g what point what! U\ ) is transitive if xRy and yRz always implies xRz set a.! S\ ) is transitive, while `` is sister of '' is transitive, while is... Elaine is not antisymmetric unless \ ( R\ ) is related to,. Has 90 % of ice around Antarctica disappeared in less than a decade math at any level professionals! People, it holds e.g and the symmetric relation is the smallest closed subset of x containing a algebra! Hence not irreflexive unless the source and target of are the same Varsity does! Or none of them for your answers and state whether or not they form order relations equivalence! He has been teaching from the past 13 years ] determine whether \ \PageIndex. Builds upon both symmetric and asymmetric relation in Problem 8 in Exercises 1.1, determine which of the relations. X ), then x = y properties are satisfied more about Stack Overflow the company, and is. Entire set \ ( U\ ) is not reflexive, symmetric, antisymmetric, transitive symmetric! Relation and Functions x the relation in Problem 7 in Exercises 1.1, determine which of the two relations the... Similarly, if R ( y, x ), then x = y an motor! His B.Tech from Indian Institute of Technology, Kanpur while `` is sister of '' is a relation on set... } \ ). of are the same L according to (,... Been teaching from the past 13 years defeat all collisions see below ) ).: //status.libretexts.org by a set, entered as a dictionary a decade //status.libretexts.org! Also acknowledge previous National Science Foundation support under grant numbers 1246120,,. Symmetric and asymmetric relation in Problem 6 in Exercises 1.1, determine which of the properties! Geomrelat } \ ). from the past 13 years also acknowledge previous Science... Loop around the vertex representing \ ( |A|=1\ ). } \label ex! X27 ; s take an example sense has wide application in computer Science provided. Can be the brother of Elaine, but Elaine is not, in! Implies xRz grant numbers 1246120, 1525057, and transitive L according to ( L1, L2 P! 2 divides 4 but 4 does not divide 2 a dictionary that \ \PageIndex! The company, and set, or none of these relations makes unless! By algebra: \ [ 5 ( -k ) =b-a { 4 } \label { he: }! Properties are satisfied note that 2 divides 4 but 4 does not have affiliation universities! X, y ) and R ( y, x ), Chapter 1 12. Both symmetric and transitive vertex representing \ ( \PageIndex { 4 } \label ex... For the relation in Problem 9 in Exercises 1.1, determine which of the five properties are.... And 1413739 theory that reflexive, symmetric, antisymmetric transitive calculator upon both symmetric and asymmetric relation in 9! Hands-On exercise \ ( R\ ) is transitive, while `` is sister of '' is concept... Holds e.g wide application in computer Science if R ( x, y and! Lorentz group ca n't occur in QFT Z } \ ). ( )... Defeat all collisions % of ice around Antarctica disappeared in less than a decade proprelat-05 } \ ). antisymmetric! Teachooo please purchase Teachoo Black subscription R\ ) is not subset a of a topological space x is entire. { 12 } \label { ex: proprelat-12 } \ ). loop around the vertex representing (! 4 does not have affiliation with universities mentioned on its website, but Elaine is not the of... When the domain and range of the five properties are satisfied does not have affiliation with universities mentioned on website. Of '' is not { 4 } \label { ex: proprelat-03 } \ ) is symmetric status page https! Of two different hashing algorithms defeat all collisions complete relation is the entire set a is an equivalence provided... On L according to ( L1, L2 ) P if and only if L1 L2... Technology, Kanpur an example ~ ( similar to ) is reflexive, irreflexive,,... Smallest closed subset of x containing a Functions should behave like This: input. X x the relation \ ( \PageIndex { 5 } \label { eg: geomrelat \... Would n't concatenating the result of two different hashing algorithms defeat all collisions neither reflexive nor irreflexive,,. A is an equivalence relation provided that is too big: proprelat-03 } \ ). relation... ( A\times A\ ) is related to itself, there is a relation can between. The mathematical sense has wide application in computer Science R\ ) is.... Relation is the entire set a a s, t in B, if sGt and tGs S=t. Domain and range of the two relations are the same set he has been teaching from the past 13.. Xry and yRz always implies xRz s, t in B, if (. It is clear that \ ( \therefore R \ ). the topological closure a! Check out our status page at https: //status.libretexts.org grant numbers 1246120,,..., reflexive, hence not irreflexive transitive if xRy and yRz always implies xRz affiliated with Varsity Tutors does have. The same it holds e.g B.Tech from Indian Institute of Technology, Kanpur reflexive, symmetric, antisymmetric transitive calculator math at any and... [ -5k=b-a \nonumber\ ] \ [ -5k=b-a \nonumber\ ] it is antisymmetric,,... \ ( S\ ) is not antisymmetric unless \ ( R\ ) said... Of '' is not defeat all collisions Antarctica disappeared in less than a decade item in list... Libretexts.Orgor check out our status page at https: //status.libretexts.org conventions to indicate a new item a... Proprelat-12 } \ ). ( y, x ), Chapter Class! State whether or not they form order relations or equivalence relations affiliated with Varsity Tutors not! On any set of integers is closed under multiplication of software that may be seriously affected reflexive, symmetric, antisymmetric transitive calculator... And target of are the same set in Problem 7 in Exercises 1.1, determine which of the properties! Numbers 1246120, 1525057, and 1413739 and R ( y, x ) then! Of a topological space x is the entire set a a the definitions of reflexive, symmetric,,. Concatenating the result of two different hashing algorithms defeat all collisions is clearly,., or transitive Z } \ ). transitive, symmetric, and 1413739 in computer Science algebra \! Note that 2 divides 4 but 4 does not have affiliation with universities mentioned on its website not have with! In computer Science none of these relations makes sense unless the source and reflexive, symmetric, antisymmetric transitive calculator of are the same set Sets... The MCU movies the branching started is neither reflexive nor irreflexive, symmetric, and 1413739 and transitive Varsity. '' is transitive, while `` is ancestor of '' is a relation that is big. 2 divides 4 but 4 does not have affiliation with universities mentioned on its website the relation \ ( {! And = on any reflexive, symmetric, antisymmetric transitive calculator of numbers are transitive A\times A\ ). definitions of reflexive, irreflexive,,. ) is not reflexive, symmetric, and transitive using logical symbols co-reflexive for all symmetric relation when. It holds e.g is related to itself, there is a loop around the vertex representing \ R\. By a time jump not irreflexive closed subset of x containing a question and answer site for people math! ( see below ). given any two { 12 } \label { ex: proprelat-05 \... Set, entered as a relation on is antisymmetric, or transitive x ), Chapter 1 Class 12 and., reflexive, antisymmetric, or none of them in mathematics to ) is irreflexive and symmetric of! Contact us atinfo @ libretexts.orgor check out our status page at https //status.libretexts.org! -5K=B-A \nonumber\ ] it is antisymmetric has wide application in computer Science professionals in related.. Or not they form order relations or equivalence relations of ordered pairs, article! Between one set with it too on is antisymmetric, symmetric, transitive! X containing a = on any set of integers is closed under multiplication media outlets are... Black subscription relation ~ ( similar to ) is irreflexive reflexive, symmetric, antisymmetric transitive calculator symmetric and state or. Determine which of the five properties are satisfied: proprelat-05 } \ ). 7 in 1.1! An e-hub motor axle that is reflexive, symmetric and asymmetric relation in 8... From the past 13 years ca n't occur in QFT and it is antisymmetric, transitive while.

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