Can someone give me an example for this question on reflexivity please. How is this reflexive? Can you show an example with numbers from the set?
Question 6
Let $X = \{0,1,2,3,\dotsc, 9\}$. Define a relation $\mathcal R$ from $X$ by "$x$ is related to $y$ if $x$ and $y$ gives the same remainder on division by $3$".
(a) Show that $\mathcal R$ is an equivalence relation.
Anwser:
(a) $\mathcal R$ is clearly both reflexive and symmetric, for every number has the same (unique) remainder on division by $3$ as itself and if $a$ has the same remainder on division by $3$ as does $b$, then $b$ has the same remainder on division by $3$ as $a$ also. Similarly $\mathcal R$ is transitive, for if $a\mathcal R b$ and $b\mathcal R c$, then $a$, $b$, and $c$ all have the same remainder on division by $3$, so $a\mathcal R c$ also.
The reflexive property, in many cases, is very subtle. This is where your difficulty lies. To show that an element x in the set X is reflexive, we need to show x R x.
So, does 0 R 0 ? (It may help to think of the zero on the left and zero on the right as being two different entities, that is, we are picking two elements from your set, where we can pick the same element twice.)
The answer is yes, since 0/3 = 0 and 0/3 = 0. That is, the 'left' zero and the 'right' zero have the same remainder after dividing by 3. The same argument applies for the rest of {1,...,9}.
Now, this has nothing to do with reflexivity, but may help in understanding the relation R.
Does 3R9? Well, 3/3 = 1 (remainder 0) and 9/3 = 3 (remainder 0), so yes, 3R9.
Does 4R9? Well, 4/3 = 1 + 1/3 (remainder 1) and 9/3 = 3 (remainder 0), so 4 "notR" 9.