Let C be the set of courses. Define the following binary relations E and P on th

Question

Let C be the set of courses. Define the following binary relations E and P on the set C: E is the relation on C where xEy means that course x is equivalent to course y; i.e. they have the same number of credits and they cover similar material.. P is the relation on C where xPy means that course x is co-requisite to course y; i.e. course x must be taken in the same semester with course y or x must completed before course y. a. Circle all the properties that E satisfies: Reflexive Symmetric Anti-symmetric Transitive Equivalence Partial Order Total Order b. Circle all the properties that P satisfies: Reflexive Symmetric Anti-symmetric Transitive Equivalence Partial Order Total Order

Explanation / Answer

Hey, To solve this, we need to understand the definitions of the properties mentioned in the options,

let me explains each one in detail:

1. Reflexive: A relation in which all the elements follow the property AA. i.e.
All the elements are related to themselves is known as reflexive relation.

So, in this case E will be a reflexive relation because every course will have same content as itself with same material.also P will also be a reflexive relation as every course in C will be taken in same semester as itself.

2. Symmetric: A relation in which all the elements follow the property such that, if AB then BA is said to be Symmetric relation.

So, in this case E will be a symmetric relation because if course x is same as y that also means y is same as x.
However, P will not be symmetric because if x is a co-requisite to y i.e x must be completed before y (A->B)

but, (B->A) i.e y is a co-requisite of x is not gauranteed and not valid.

3. Transitive:A relation in which all the elements follow the property such that, if AB and BC then AC is said to be Transitive relation.

In this case, E will be transitive because if a course x is similar to y and course y is similar to z i.e

number of credits of x= number of credits of y= number of credits of z and also the content, hence transitive.

Now, for P. consider 3 courses x,y,z. x is a co-requisite of y and y is a co-requisite of z i.e to complete z we need to complete y, but to complete y we need to complete x thereby to take z, x needs to be completed. which makes P transitive.

4.Equivalence: A relation which has all the above 3 properties, that is, if its reflexive, symmetric and transitive.

therefore, E is an equivalence relation and P is not.

5. Anti-Symmetric: Its the opposite of symmetric i.e if R(a,b) with a b, then R(b,a) must not hold.

E has already been established as Symmetric, lets decide on P.

Considet 2 courses x and y. If x is a prerequisite of y then for y to beprerequisite of x only case existe is x=y. therefore P is anti symmetric.

6. Parial Order:R is a partial order relation if R is reflexive, antisymmetric and transitive.

Since E is Symmetric it is not a partial order relation.And P will be a partial order as it is all 3 as explained above.

7. Total Order: A relation is of Total Order if its a Partial order and every element is related to each other in the relation.

E is not a partial order, hence not total order.

Now, P consider courses x,y,z,a,b,c.

x is co-requisite of y, a is of b, z is of c. Now there could be courses in C such that they are not co-requistes of each other which is possible, hence P is NOT a total order relation.

Hence the final solution,

E: Reflexive,Symmetric,Transitive, Equivalence.

P: Reflexive,Anti-Symmetric,Transitive, Partial Order

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.