How to write $x$ is not related to $y$ in binary relations?

Question

I am trying to prove a relation is not transivitive. So I start with $xRy$ and $yRz$. Then I want to write: "But, $x$ doesn't relate to $z$." How do I write that mathematically?

Answer 1

An answer combining the comments:

You could write:

• $(x,y)\in R$ and $(y,z)\in R$, but $(x,z)\notin R$.

• $xRy$ and $yRz$, but $\neg (xRz)$.

• $xRy$ and $yRz$, but $\require{cancel} x \cancel{R} y$.

• $x$ relates to $y$ and $y$ relates to $z$, but $x$ does not relate to $z$.

I added that last one because you ended your question with "How do I write that mathematically?" To write something mathematically does not mean use crazy symbols. If something is written clearly, precisely and unambiguously, then it is written mathematically.

The only reason to use symbols is to save space, which can make things easier to read (though not necessarily). The validity of a statement does not change if you choose to use words in stead of symbols.

Answer 2

Notice that $R$ is a set of ordered pairs $(x,y)$ such that x is related to $y$

Thus $$(x,y) \in R \text { and } (y,z) \in R, \text { but } (x,z) \notin R$$ is a mathematically correct way to express your idea.