Reflexive, Symmetric or Transitive

Question

Here are the questions

for R3:

    Relexive because (1,1) and 1/1 =1

    Not Symmetric because i = 123 j = 7 then 123/7 >= 0 but not 7/123

    Transitive because I>= J>= K

R4:

Not Reflexive because floor pi equal floor pi  x = pi, x = x, pi = pi, 4 = 4

Completly stuck in reflexive and transtive

If i am incorrect can you tell me why the answer is what it is?

Also with these relation how do i have a generalization to prove that it is reflexive or symmetric or transitive?

Thanks in advance!

Answer 1

For your question in $R4,$ the reflexity must hold for given $x\in \mathbb{R},$ we always have $\lfloor x\rfloor = \lfloor x\rfloor.$

There is a general and only way to check such equivalence relation that is to follow the definition of you relations, and don't be shy to discuss with others!