4) Let R be a relation on the set [1,2,3,., 8,9,10). aRb is in the relation if a

Question

4) Let R be a relation on the set [1,2,3,., 8,9,10). aRb is in the relation if a and b are distinct integers and there is a common divisor that is not 1 shared between a and b. 2R4, 6R3, and 10R2 are all members of the relation, while 5R2 and 7R9 are not. a) Is it possible to draw a reduced graph for this relation? Why or why not? b) What if we change the condition to be that of a cleanly dividing into b (no remainder)? Can our new relation be drawn as a reduced graph? Why or why not? c) Let us assume we have all pairs of our original relation enumerated, and no knowledge of t relation function (so all we know is the current pair list). What pairs would we have to add to them in order to change R into a partial order relation? Into an equivalence relation?

Explanation / Answer

c: we know that the relation ispartial order relation if it is a. reflexive , b. antisymmetric,c. transitive.

so here 2R2....., 2R4(Excluding 4R2),....... 2R4, 4R8,2R8.... SUCH PAIRS CAN BE ADDED ON TO MAKE THE RELATION A PARTIALORDERED.

Now tomake the relation a equivalance one ... it should be a. reflexive, b. symmetric, c. transitive.

so for making it symmentric we can add on the pairs like 2R4, 4R2... and so on... rest of the pairs can be used in the similar way as done above for reflexive and transitive.

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