Let X be the set of all students registered in AUT in 2016 F

Let X be the set of all students registered in AUT in 2016. For each of the following relations defined on X, carefully explain whether or not it is reflexive, symmetric, anti-symmetric, and/or transitive. R_1 = {(x, y) epsilon P^2 |x has a common course with y}, where P is a set of students. R_2 = {(x, y) epsilon P^2 |x has exactly the same number of letters in the name as y}, where P is a set of students.

Solution

( i ) The relation R₁ is defined by R₁ = { ( x , y ) P² | x has a common course with y }

( ii )The relation R₂ is defined by R₂ = { ( x , y ) P² | x has exactly the same number of letters in the name as y }

1. Since every student has exactly the same number of letters in the name as himself.

That is x P , (x , x ) R₂. So R₂ is reflexive.

2. Suppose x has exactly the same number of letters in the name as y then y has exactly the same number of letters in the name as x.

That is if ( x , y ) R₂ then( y , x ) R₂. So, R₂ is symmetric.

3. Suppose x has exactly the same number of letters in the name as y and y has exactly the same number of letters in the name as x.Then x and y may not same.

That is if ( x , y ) R₂ and ( y , x ) R₂ then x may not be equal to y.So, R₂ is not anti-symmetric.

4. Suppose x has exactly the same number of letters in the name as y and y has exactly the same number of letters in the name as z.Then x has exactly the same number of letters in the name as z.

That is if ( x , y ) R₂ and ( y , z ) R₂ then ( x , z ) R₂ .So, R₂ is transitive.