Problem 1.3.For a b in R define aRb to mean that |a-b|<5. Prove or disprove each of the following: 1.The relation R is reflexive. 2.The relation R is symmetric. 3.The relation R is transitive.

Hello Siddhant!

This question is dependent on the values in the set R, which you have not mentioned in the question. However, I shall tell you the method to solve the problem.

In the set R, find all the pairs of numbers which holds the relation |a-b| < 5. For example, if R={1,2,8,5} then |a-b|<5 are {(1,2),(2,1),(5,8),(8,5),(1,5),(5,1),(1,1),(2,2),(8,8),(5,5)}. Note that since there is modulus operator, even the negative difference will be converted to positive and the difference must be strictly less than 5.

Now, based on the above relation and the set we obtained, answer the given questions.

The relation R is said to be reflexive if for every element a in R, (a,a) belongs to S where S is the set obtained for relation aRb. As you can see for the above example it does holds good. Every number sbtracted from itself gives 0 which is less than 5. So the relation is reflexive.

The relation is said to be symmetric if (a,b) belongs to S then (b,a) belongs to S. As you can see, for our example, it is symmetric as well.

If (a,b) is in S, (b,c) is in S then (a,c) is also is in S. This is transitive relation. which is satisfied in our example.

Hence, in the example I have taken the relation is reflexive, symmetric and transitive. Apply the same logic to the given set R and find out the answers.