. Union – consists of all ordered pairs from both relations. Reflexive Closure – is the diagonal relation on set. The reflexive closure of relation on set is. Symmetric Closure – Let be a relation on set, and let be the inverse of. The symmetric closure of relation on set is. Transitive Closure – Let be a relation on set.

The connectivity relation is defined as –. The transitive closure of is.Example – Let be a relation on set with. Find the reflexive, symmetric, and transitive closure of R.Solution –For the given set,. So the reflexive closure of isFor the symmetric closure we need the inverse of, which is.The symmetric closure of is-For the transitive closure, we need to find.we need to find until.

Closure Property Addition. Don't forget to try our free app - Agile Log, which helps you track your time spent. What algebraic property does this statement show? 3 + (–7) = (–7) + 3 (Points: 5) associative property commutative property closure property symmetric property Question 2.2. 3x – 5 = 7x – 17 (Points: 5) 2 3 5 6 Question 3.3. What is the slope of the line that passes through the points (3, –1) and (–2, –5)? (Points: 5) Question 4.4. The sum of two consecutive odd.

We stop when this condition is achieved since finding higher powers of would be the same.Since, we stop the process.Transitive closure, –Equivalence Relations:Let be a relation on set. If is reflexive, symmetric, and transitive then it is said to be a equivalence relation.Consequently, two elements and related by an equivalence relation are said to be equivalent.Example – Show that the relationis an equivalence relation. Is the congruence modulo function. It is true if and only if divides.Solution – To show that the relation is an equivalence relation we must prove that the relation is reflexive, symmetric and transitive. Reflexive – For any element, is divisible. So, congruence modulo is reflexive. Symmetric – For any two elements and, if or i.e.

Is divisible by, then is also divisible. So Congruence Modulo is symmetric.

Transitive – For any three elements, and if then-Adding both equations. So, is transitive.Since the relation is reflexive, symmetric, and transitive, we conclude that is an equivalence relation.Equivalence Classes:Let be an equivalence relation on set.We know that if then and are said to be equivalent with respect to.The set of all elements that are related to an element of is called theequivalence class of.

It is denoted by or simply if there is only onerelation to consider.Formally,Any element is said to be the representative of.Important Note: All the equivalence classes of a Relation on set are either equal or disjoint and their union gives the set.The equivalence classes are also called partitions since they are disjoint and their union gives the set on which the relation is defined. Example: What are the equivalence classes of the relation Congruence Modulo?. Solution: Let and be two numbers such that.

Dead secretary costume kids. This means that the remainder obtained by dividing and with is the same.Possible values for the remainder-Therefore, there are equivalence classes –GATE CS Corner QuestionsPracticing the following questions will help you test your knowledge. All questions have been asked in GATE in previous years or in GATE Mock Tests. It is highly recommended that you practice them.1.2.3.4.References –Discrete Mathematics and its Applications, by Kenneth H RosenThis article is contributed by Chirag Manwani. If you like GeeksforGeeks and would like to contribute, you can also write an article using or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.