Activity 10- non graded

[Bhumika Taneja]

how is this relation transitive and please explain the other question too?

[Varun Vankayala]

First question:
Transitive property states that if (a,b) ∈ R and (b,c) ∈ R, then (a,c) also  ∈ R.
We start with picking out sets of two elements (a,b) and (b,c) such that b exists in both, as second part of a pair in one, and as first part of a pair in the other. When we find such pairs, we look for another element (a,c). If such an element (a,c) exists for all existing cases of (a,b),(b,c) in R, we can say that R is transitive.
Let us start with the first element (Snail,Frog). We look for any pairs that start with (Frog,_____). There are none.
Second element (Bird,Bird) is of form a,a and is therefore skipped.
Third element (Fox, Frog), we look for pairs starting with (Frog,_____). There are none.
Final element (Snail,Fox). We look for pairs starting with (Fox,___), and there is one: (Fox,Frog). Therefore, for the relation to be transitive, (Snail,Frog) must exist, which it does in element 1. Therefore R is transitive.

Second question:
The question essentially is asking you how many pairs or elements need to be added to R, for R to become symmetric.
Symmetry states that if (a,b) ∈ R, then (b,a) must also  ∈ R.

So essentially we must add elements to R such that there is a reverse existing for it. Then we count how many we needed to add, and that is the answer.  Go element by element, see if its reverse already exists in R, and if it doesn't, add the reverse. Repeat for all elements, and see how many you had to add.