Consider a (binary) relationRon a non-empty setSconsisting of points in a plane. Letaandbbe twoelements ofS, i.e.,a,b?S, and defineRsuch thata R biff there is a directed path fromatob. Consider the...

Consider a (binary) relationRon a non-empty setSconsisting of points in a plane. Letaandbbe two elements ofS, i.e.,a,b?S, and defineRsuch thata R biff there is a directed path fromatob.

1. 
   

   Consider the setS={a,b,c}with(a,b),(b,c),(c,d)?R1. IsR1transitive? Show this.
   

   
   


2. 
   

   Consider the setS={a,b,c}with(a,b),(b,c),(c,b),(c,d),(a,c),(b,d)?R2. IsR2transitive? Show
   

   
   

   this.
   

   
   


3. 
   

   Define the transitive closure ofRtofR.
   

   
   


4. 
   

   Construct the transitive closure ofR1.
   

   
   


5. 
   

   Provide a concise description of the method (algorithm) you used to construct the transitive closure ofR1.
   

   
   


6. 
   

   Givennpoints in the plane, what is the maximum number of distinct directed paths that can connect these points. (This problem requires you to carefully define the structure of the points in the plane).