Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. You can test out of the How Do I Use Study.com's Assign Lesson Feature? So are closed paths and closed balls. Each wheel is a closed set because you can't go outside its boundary. We will now look at some examples of the closure of a set The set is not completely bounded with a boundary or limit. Rowland. Take a look at this set. Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Let's see. . The closure of the open 3-ball is the open 3-ball plus the surface. . The digraph of the transitive closure of a relation is obtained from the digraph of the relation by adding for each directed path the arc that shunts the path if one is already not there. Sciences, Culinary Arts and Personal Get the unbiased info you need to find the right school. If you take this approach, having many simple code examples are extremely helpful because I can find answers to these questions very easily. in a nonempty set. The interior of G, denoted int Gor G , is the union of all open subsets of G, and the closure of G, denoted cl Gor G, is the intersection of all closed Consider a sphere in 3 dimensions. Quiz & Worksheet - What is a Closed Set in Math? first two years of college and save thousands off your degree. In general, a point set may be open, closed and neither open nor closed. Math has a way of explaining a lot of things. If attribute closure of an attribute set contains all attributes of relation, the attribute set will be super key of the relation. 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Is X closed? For example, the set of real numbers, for example, has closure when it comes to addition since adding any two real numbers will always give you another real number. One might be tempted to ask whether the closure of an open ball. However, when I check the closure set $(0, \frac{1}{2}]$ against the Theorem 17.5, which gives a sufficient and necessary condition of closure, I am confused with the point $0 \in \mathbb{R}$. Examples… Closure of a Set of Functional Dependencies. For example, the set of even natural numbers, [2, 4, 6, 8, . Anything that is fully bounded with a boundary or limit is a closed set. Interior, Closure, Exterior and Boundary Let (X;d) be a metric space and A ˆX. Look at this fence here. If F is used to donate the set of FDs for relation R, then a closure of a set of FDs implied by F is denoted by F +. I don't like reading thick O'Reilly books when I start learning new programming languages. Now, We will calculate the closure of all the attributes present in … The transitive closure of is . Example- In the above example, The closure of attribute A is the entire relation schema. Explore anything with the first computational knowledge engine. {{courseNav.course.mDynamicIntFields.lessonCount}} lessons It is useful to be able to distinguish between the interior of 3-ball and the surface, so we distinguish between the open 3-ball, and the closed 3-ball - the closure of the 3-ball. Earn Transferable Credit & Get your Degree. Closure of Attribute Sets Up: Functional Dependencies Previous: Basic Concepts. © copyright 2003-2020 Study.com. • In topology and related branches, the relevant operation is taking limits. In topology, a closed set is a set whose complement is open. Example 3 The Closure of a Set in a Topological Space Examples 1 Recall from The Closure of a Set in a Topological Space page that if is a topological space and then the closure of is the smallest closed set containing. 2. The class will be conducted in English and the notes will be provided in English. credit by exam that is accepted by over 1,500 colleges and universities. Arguments x. Unfortunately the answer is no in general. https://mathworld.wolfram.com/SetClosure.html. Typically, it is just A with all of its accumulation points. To learn more, visit our Earning Credit Page. Find the reflexive, symmetric, and transitive closure of R. Solution – For the given set, . Examples: The transitive closure of a parent-child relation is the ancestor-descendant relation as mentioned above, and that of the less-than relation on I is the less-than relation itself. Candidate Key- If there exists no subset of an attribute set whose closure contains all the attributes of the relation, then that attribute set is called as a candidate key of that relation. Hence, result = A. For binary_closure and binary_reduction: a binary matrix.A set of (g)sets otherwise. Any operation satisfying 1), 2), 3), and 4) is called a closure operation. It sets the counter to zero (0), and returns a function expression. Closure of a Set 1 1.8.6. Many topological properties which are defined in terms of open sets (including continuity) can be defined in terms of closed sets as well. ], is closed with respect to addition because the sum of any two of them is another even natural number, which is also a member of the set. Or, equivalently, the closure of solid S contains all points that are not in the exterior of S. Examples Here is an example in the plane. You'll learn about the defining characteristic of closed sets and you'll see some examples. 4. Lesson closure is so important for learning and is a cognitive process that each student must "go through" to wrap up learning. Example – Let be a relation on set with . This definition probably doesn't help. Let us discuss this algorithm with an example; Assume a relation schema R = (A, B, C) with the set of functional dependencies F = {A → B, B → C}. Example: The set {1,2,3,4,5} has no boundary points when viewed as a subset of the integers; on the other hand, when viewed as a subset of R, every element of the set is a boundary point. You can also picture a closed set with the help of a fence. Example of Kleene star applied to the empty set: ∅* = {ε}. This example illustrates the use of the transitive closure algorithm on the directed graph G shown in Figure 19. How to use closure in a sentence. From MathWorld--A Wolfram So, you can look at it in a different way. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. The inside of the fence represents your closed set as you can only choose the things inside the fence. However, developing a strong closure, which is the fifth step in writing a strong and effective eight-step lesson plan for elementary school students, is the key to classroom success. Closure definition is - an act of closing : the condition of being closed. Amy has a master's degree in secondary education and has taught math at a public charter high school. operation. Example. Examples. The closure of a set is the smallest closed set containing {{courseNav.course.topics.length}} chapters | This can happen only if the present state have epsilon transition to other state. \begin{align} \quad [0, 1]^c = \underbrace{(-\infty, 0)}_{\in \tau} \cup \underbrace{(1, \infty)}_{\in \tau} \in \tau \end{align} Convex Optimization 6 You should change all open balls to open disks. What's the syntax for if and else? The closure of a solid S is defined to be the union of S's interior and boundary, written as closure(S). Portions of this entry contributed by Todd My argument is as follows: The unique smallest closed set containing the given Analysis (cont) 1.8. Knowledge-based programming for everyone. Mathematical examples of closed sets include closed intervals, closed paths, and closed balls. If no subset of this attribute set can functionally determine all attributes of the relation, the set will be candidate key as well. . Thus, a set either has or lacks closure with respect to a given operation. For the operation "rip", a small rip may be OK, but a shirt ripped in half ceases to be a shirt! Not sure what college you want to attend yet? Get access risk-free for 30 days, This approach is taken in . courses that prepare you to earn In other words, X + represents a set of attributes that are functionally determined by X based on F. And, X + is called the Closure of X under F. All such sets of X +, in combine, Form a closure of F. Algorithm : Determining X +, the closure of X under F. A set that has closure is not always a closed set. Hereditarily finite set. flashcard set{{course.flashcardSetCoun > 1 ? Closed sets are closed under arbitrary intersection, so it is also the intersection of all closed sets containing A. Example – Let be a relation on set with . Walk through homework problems step-by-step from beginning to end. Determine the set X + of all attributes that are dependent on X, as given in above example. A closed set is a different thing than closure. The symmetric closure … Interior, Closure, Exterior and Boundary Let (X;d) be a metric space and A ˆX. The closure of a point set S consists of S together with all its limit points i.e. For example let (X;T) be a space with the antidiscrete topology T = {X;?Any sequence {x n}⊆X converges to any point y∈Xsince the only open neighborhood of yis whole space X, and x Example Explained. Thus, a set either has or lacks closure with respect to a given operation. Example: Let A be the segment [,) ∈, The point = is not in , but it is a point of closure: Let = −. Well, definition. just create an account. It's a round fence. So shirts are closed under the operation "wash". Transitive Closure – Let be a relation on set . The complement of the interior of the complement When a set has closure, it means that when you perform a certain operation such as addition with items inside the set, you'll always get an answer inside the same set. In math, its definition is that it is a complement of an open set. However, the set of real numbers is not a closed set as the real numbers can go on to infinity. How can I define a function? In topologies where the T2-separation axiom is assumed, the closure of a finite set is itself. As a consequence closed sets in the Zariski topology are the whole space R and all ﬁnite subsets of R. 5.4 Example. of the set. . Both of these sets are open, so that means this set is a closed set since its complement is an open set, or in this case, two open sets. study That is, a set is closed with respect to that operation if the operation can always be completed with elements in the set. It has a boundary. All rights reserved. Enrolling in a course lets you earn progress by passing quizzes and exams. After reading this lesson, you'll see how both the theoretical definition of a closed set and its real world application. But, if you think of just the numbers from 0 to 9, then that's a closed set. What constitutes the boundary of X? Closure is an opportunity for formative assessment and helps the instructor decide: 1. if additional practice is needed 2. whether you need to re-teach 3. whether you can move on to the next part of the lesson Closure comes in the form of information from students about what they learned during the class; for example, a restatement of the A set that has closure is not always a closed set. Example 7. Topological spaces that do not have this property, like in this and the previous example, are pretty ugly. accumulation points. Problems in Geometry. Topological spaces that do not have this property, like in this and the previous example, are pretty ugly. Closure is based on a particular mathematical operation conducted with the elements in a designated set of numbers. These are very basic questions, but enough to start hacking with the new langu… … Hints help you try the next step on your own. One can define a topological space by means of a closure operation: The closed sets are to be those sets that equal their own closure (cf. De–nition Theclosureof A, denoted A , is the smallest closed set containing A That is, a set is closed with respect to that operation if the operation can always be completed with elements in the set. Closure of a Set • Every set is always contained in its closure, i.e. The reduction of a set \(S\) under some operation \(OP\) is the minimal subset of \(S\) having the same closure than \(S\) under \(OP\). Deﬁnition: Let A ⊂ X. What scopes of variables are available? Now, we can find the attribute closure of attribute A as follows; Step 1: We start with the attribute in question as the initial result. Closed sets are closed If you look at a combination lock for example, each wheel only has the digit 0 to 9. This is a set whose transitive closure is finite. is equal to the corresponding closed ball. $B (a, r)$. closed set containing Gis \at least as large" as G. We call Gthe closure of G, also denoted cl G. The following de nition summarizes Examples 5 and 6: De nition: Let Gbe a subset of (X;d). The, the final transactions are: x --- > w wz --- > y y --- > xz Conclusion: In this article, we have learned how to use closure set of attribute and how to reduce the set of the attribute in functional dependency for less wastage of attributes with an example. The variable add is assigned to the return value of a self-invoking function. It has its own prescribed limit. Symmetric Closure – Let be a relation on set , and let be the inverse of . Is this a closed or open set? It is also referred as a Complete set of FDs. Select a subject to preview related courses: There are many mathematical things that are closed sets. which is itself a member of . How to find Candidate Keys and Super Keys using Attribute Closure? armstrongs axioms explained, example exercise for finding closure of an attribute Advanced Database Management System - Tutorials and Notes: Closure of Set of Functional Dependencies - Example Notes, tutorials, questions, solved exercises, online quizzes, MCQs and more on DBMS, Advanced DBMS, Data Structures, Operating Systems, Natural Language Processing etc. In other words, every set is its own closure. A closed set is a different thing than closure. So shirts are not closed under the operation "rip". x 1 x 2 y X U 5.12 Note. An open set, on the other hand, doesn't have a limit. Practice online or make a printable study sheet. References We can decide whether an attribute (or set of attributes) of any table is a key for that table or not by identifying the attribute or set of attributes’ closure. Create your account, Already registered? I can follow the example in this presentation, that is to say, by Theorem 17.4, … For the operation "wash", the shirt is still a shirt after washing. Formal math definition: Given a set of functional dependencies, F, and a set of attributes X. Did you know… We have over 220 college The closure of A in X, denoted cl(A) or A¯ in X is the intersection of all Log in or sign up to add this lesson to a Custom Course. The transitive closure of is . very weak example of what is called a \separation property". I thought that U closure=[0,2] c) Give an example of a set S of real numbers such that if U is the set of interior points of S, then U closure DOES NOT equal S closure This one I was not sure about, but here is my example: S=(0,3)U(5,6) S closure=[0,3]U[5,6] 's' : ''}}. How to use closure in a sentence. b) Given that U is the set of interior points of S, evaluate U closure. As teachers sometimes we forget that when students leave our room they step out into another world - sometimes of chaos. https://mathworld.wolfram.com/SetClosure.html. Anyone can earn The closure is essentially the full set of attributes that can be determined from a set of known attributes, for a given database, using its functional dependencies. Topology of Rn (cont) 1.8.5. Closed sets, closures, and density 3.3. Shall be proved by almost pure algebraic means. Rather, I like starting by writing small and dirty code. If a ⊆ b then (Closure of a) ⊆ (Closure of b). Theorem 2.1. FD1 : Roll_No Name, Marks. De–nition Theinteriorof A, denoted intA, is the largest open set contained in A (alternatively, the union of all open sets contained in A). The connectivity relation is defined as – . Figure 11 contains various sets. A set and a binary You can think of a closed set as a set that has its own prescribed limits. We need to consider all functional dependencies that hold. However, the set of real numbers is not a closed set as the real numbers can go on to infini… The Closure Property states that when you perform an operation (such as addition, multiplication, etc.) Some are closed, some not, as indicated. So members of the set … For example, the set of real numbers, for example, has closure when it comes to addition since adding any two real numbers will always give you another real number. 5. The topological closure of a set is the corresponding closure operator. Closure relation). So the reflexive closure of is . The Kuratowski closure axioms characterize this operator. Compact Sets 3 1.9. Source for information on Closure Property: The Gale Encyclopedia of Science dictionary. Example 1: Simple Closure let simpleClosure = { } simpleClosure() In the above syntax, we have declared a simple closure { } that takes no parameters, contains no statements and does not return a value. under arbitrary intersection, so it is also the intersection of all closed sets containing A set S and a binary operator * are said to exhibit closure if applying the binary operator to two elements S returns a value which is itself a member of S. The closure of a set A is the smallest closed set containing A. This class would be helpful for the aspirants preparing for the IIT JAM exam. Create an account to start this course today. Think of it as having a fence around it. For the symmetric closure we need the inverse of , which is. A ⊆ A ¯ • The closure of a set by definition (the intersection of a closed set is always a closed set). Closure is an opportunity for formative assessment and helps the instructor decide: 1. if additional practice is needed 2. whether you need to re-teach 3. whether you can move on to the next part of the lesson Closure comes in the form of information from students about what they learned during the class; for example, a restatement of the If you picked the inside, then you are absolutely correct! Examples: The transitive closure of a parent-child relation is the ancestor-descendant relation as mentioned above, and that of the less-than relation on I is the less-than relation itself. IfXis a topological space with the discrete topology then every subsetA⊆Xis closed inXsince every setXrAis open inX. Boundary of a Set 1 1.8.7. The closure of a set \(S\) under some operation \(OP\) contains all elements of \(S\), and the results of \(OP\) applied to all element pairs of \(S\). 1.8.5. . The complement of this set are these two sets. The set of identified functional dependencies play a vital role in finding the key for the relation. . The reflexive closure of relation on set is . $\bar {B} (a, r)$. When a set has closure, it means that when you perform a certain operation such as addition with items inside the set, you'll always get an answer inside the same set. The set of all those attributes which can be functionally determined from an attribute set is called as a closure of that attribute set. The term "closure" is also used to refer to a "closed" version of a given set. An algebraic closure of K is a field L, which is algebraically closed and algebraic over K. So Theorem 2, any field K has an algebraic closure. . Theorem: A set A ⊂ X is closed in X iﬀ A contains all of its boundary points. Closure of a Set. equivalent ways, including, 1. Example of Kleene plus applied to the empty set: ∅+ = ∅∅* = { } = ∅, where concatenation is an associative and non commutative product, sharing these properties with the Cartesian product of sets. Closed sets We will see later in the course that the property \singletons are their own closures" is a very weak example of what is called a \separation property". De–nition Theinteriorof A, denoted intA, is the largest open set contained in A (alternatively, the union of all open sets contained in A). Closure of a set. Example: the set of shirts. Example-1 : Consider the table student_details having (Roll_No, Name,Marks, Location) as the attributes and having two functional dependencies. . . As teachers sometimes we forget that when students leave our room they step out into another world - sometimes of chaos. Figure 12 shows some sets and their closures. Log in here for access. If attribute closure of an attribute set contains all attributes of relation, the attribute set will be super key of the relation. The set plus its limit points, also called "boundary" points, the union of which is also called the "frontier.". Example- I have having trouble with some simple problems involving the closure of sets. Web Resource. But if you are outside the fence, then you have an open set. This set is formed from the values of all finite sequences x 1, …, x h (h integer) such that x 1 ∈ a and x i+1 ∈ x i for each i(1 ≤ i < h). The Bolzano-Weierstrass Theorem 4 1. Implicitly there are two regions of interest created by this sphere; the sphere itself and its interior (which is called an open 3-ball). It is so close, that we can find a sequence in the set that converges to any point of closure of the set. People can exercise their horses in there or have a party inside. A closed set is a set whose complement is an open set. Join the initiative for modernizing math education. You can't choose any other number from those wheels. If no subset of this attribute set can functionally determine all attributes of the relation, the set will be candidate key as well. FD2 : Name Marks, Location. The connectivity relation is defined as – . Figure 19: A Directed Graph G The directed graph G can be represented by the following links data set, LinkSetIn : Here's an example: Example 1: The set "Candy" Lets take the set "Candy." Closed Sets 34 open neighborhood Uof ythere exists N>0 such that x n∈Ufor n>N. The set operation under which the closure or reduction shall be computed. Let's consider the set F of functional dependencies given below: F = {A -> B, B -> C, C -> D} The symmetric closure of relation on set is . and career path that can help you find the school that's right for you. When a set has closure, it means that when you perform an operation on the set, then you'll always get an answer from within the set. To unlock this lesson you must be a Study.com Member. The boundary of the set X is the set of closure points for both the set X and its complement Rn \ X, i.e., bd(X) = cl(X) ∩ cl(Rn \ X) • Example X = {x ∈ Rn | g1(x) ≤ 0,...,g m(x) ≤ 0}. Closed intervals for example are closed sets. This doesn't mean that the set is closed though. Closure Property The closure property means that a set is closed for some mathematical operation. Closure are different so now we can say that it is in the reducible form. Your numbers don't stop. Closure definition is - an act of closing : the condition of being closed. Epsilon means present state can goto other state without any input. ( such as addition, multiplication, etc. step out into another world - sometimes chaos. Functionally determine all attributes of relation, the attribute set contains all attributes of relation the. Here, our concern is only with the help of examples set under... Ways, including, 1 the original set in a different thing than closure. closure! Of attributes X Yellow Wallpaper the theoretical definition of a set is a complement of this set are these sets... This can happen only if the operation `` rip '' need the inverse of, which do. Or have a party inside your own to these questions very easily the real numbers go... Given that U is the closure of all the attributes present in … example of what is the smallest set. Trademarks and copyrights are the property of their respective owners ca n't go outside its boundary.. The Yellow Wallpaper do not have this property in Geometry a way of explaining lot... Always a closed set equivalent ways, including, 1 of, which part do you think a. An open set G the directed graph G shown in Figure 19 thick O'Reilly books when I start new. Helpful for the given set of all closed sets, closures, and closed balls mathematical operation conducted with discrete. ) is called a closure operation a designated set of shirts so now we can prove that certain other also., including, 1 satisfying 1 ), and density 3.3 have this property like. Same time H. T. ; Falconer, K. J. ; and Guy, R. K. problems. Find the reflexive, symmetric, and a set either has or lacks closure with respect a. Using attribute closure defining characteristic of closed sets 34 open neighborhood Uof ythere exists N >.. Y X U 5.12 Note complement is an open set, on the other hand, does n't a. Wash '' Springer-Verlag, p. 2, 1991 metric space and a set a. The return value of a set that converges to any point of closure of a set of numbers respect a! 2 ), 2 ), 2 ), and closed balls any point of closure an! And has taught math at a public charter high school axiom is assumed, the operation... Find a sequence may converge to many points at the same time 1 tool for creating Demonstrations and technical. Algorithm on the other hand, does n't mean that the set shirts! Binary_Reduction: a directed graph G shown in Figure 19 points of S together with all its limit i.e..., a set F of functional dependencies, we will calculate the closure of a an act of:. 3-Ball plus the surface the directed graph G the directed graph G shown in Figure 19 n't like reading O'Reilly! Encyclopedia of Science dictionary you should change all open balls to open disks dependencies... 19: a directed graph G shown in Figure 19: a binary matrix.A set of functional,... That 's an example: the set operation under which the closure of )... You have an open set by writing small and dirty code axiom assumed. In finding the key for that relation or sign up to add this lesson you must be relation! Has or lacks closure with respect to that operation if the present state have epsilon transition other. The surface other closure of a set examples also hold thus, attribute a is a complement of this set... Its limit points i.e or have a party inside a Complete set of identified functional dependencies a! However, the result of the interior of the first two years of college and save thousands your! ( 0 ), and a set • every set is its own closure ''! How both the theoretical definition of a closed set is the smallest closed as. Consider all functional dependencies play a vital role in finding the key for relation... Binary_Reduction: a binary matrix.A set of identified functional dependencies that hold example, are ugly! Fds that can be represented by the following example will … example of what is called a operation... Closed and neither open nor closed intersects the original set in a nonempty set is in the that! The next step on your own and you 'll see some examples weak example what. Absolutely correct unbiased info you need to consider all functional dependencies, we will calculate the closure of closed. The present state have epsilon transition to other state would be helpful for the ``... Then it is also the intersection of all possible FDs that can be represented the. Many mathematical things that are closed under the operation can always be completed with elements the. Can access the counter in the same set problems step-by-step from beginning to end under intersection. This in the same time operation if the present state have epsilon transition to state! Weisstein, Eric W. `` set closure of a set examples. is closed a closed set is a set its! Is its own closure. may be open, closed and neither nor... Lesson you must be a Study.com Member get access risk-free for 30 days, create! Examples are extremely helpful because I can find answers to these questions easily. Choose any other number from those wheels, LinkSetIn Tomar will discuss interior of the open 3-ball plus surface. Its boundary Complete set of ( G ) sets otherwise Candy '' lets the... Respect to that operation if the operation `` wash '', the set is open... Binary matrix.A set of even natural numbers, [ 2, 4, 6, 8.. After washing room they step out into another world - sometimes of.... N'T go outside its boundary points ⊆ ( closure of b ) need the of. '' part is that it is just a with all of its boundary and copyrights are the of... Property states that when students leave our closure of a set examples they step out into another world - sometimes of.! College you want to attend yet several equivalent ways, including, 1 after washing,... Open inX outside of the open 3-ball is the smallest closed set containing every... A given operation a X, A= a the following example will … example: 1... Fully bounded with a boundary or limit sets containing learn about the defining characteristic of closed sets closed! English and the previous example, each wheel is a closed set their horses in there or have a.... Just the numbers from 0 to 9 the computation is another number the... Operation ( such as addition, multiplication, etc. a self-invoking function condition of being closed one. Up to add this lesson you must be a Study.com Member books when I start new! For any a X, A= a 1 ), and Let a... Own prescribed limits, closure, Exterior and boundary Let ( X ; d be!, which part do you think of just the numbers from 0 to 9 leave our room they out... Key as well that each student must `` go through '' to up!, visit our Earning Credit Page operation under which the closure of finite... Closure with respect to a given set the computation is another number in the Wallpaper. When students leave our room they step out into another world - sometimes of.. To any point of closure of a fence lot of things A= a the open 3-ball is Rest... Complete set of shirts b } ( a, is the corresponding closure operator, and. References interior, closure, Exterior and boundary Let ( X ; d ) be a space! Not, as indicated whether the closure of the transitive closure algorithm on the directed graph G in. Branches, the relevant operation is taking limits for learning and is a closed set is.! Denoted a, denoted a, r ) $ this set are closure of a set examples two.!, on the directed graph G the directed graph G shown in Figure 19 after.... Closed in X iﬀ a contains all of its accumulation points it is in the set add this,... Out into another world - sometimes of chaos, then it is closed though 6.in ( X ; d be! York: Springer-Verlag, p. 2, 4, 6, 8, aspirants... Is taking limits closing: the condition of being closed closure of a set examples it can access the to... Closure of a finite set is not always a closed set is closed with respect that... Two years of college and save thousands off your degree the fence determine all attributes of relation, relevant. Plus the surface transitive closure – Let be a metric space and a.... Which is so, you 'll see some examples candidate key as well on set candidate key well. X 2 y X U 5.12 Note, that we can prove that certain ones. You include all the attributes present in … example of Kleene star applied to return... Operation `` rip '' how to find candidate Keys and super Keys using attribute closure of an open.! Relation, the set if attribute closure of a self-invoking function is always contained its... Property as it applies to real numbers intersection of all closed sets containing present in … example: example:... This and the previous example, are pretty ugly and the previous example, are pretty ugly take! A, r ) $ dirty code so now we can prove that certain ones! Mathematical things that are closed under arbitrary intersection, so it is used.