why "not equal to " does not satisfy transitivity and reflexivity.

Question

caozeyuan · Answer

@UnkleRhaukus you are the only one who stay here for more than 5 seconds

jim_thompson5910 · Answer

$\Large x \ne y$ and $\Large y \ne z$ then $\Large x \ne z$ is not true in general.


Say for example, x = 1, y = 2, and z = 1.


That would mean


$\Large x \ne y$ and $\Large y \ne z$ then $\Large x \ne z$


turns into


$\Large 1 \ne 2$ and $\Large 2 \ne 1$ then $\Large 1 \ne 1$


which is false. So that proves the "not equal to" is not transitive.

jim_thompson5910 · Answer

Oh sorry forgot my "if" words to place at the beginning. But you hopefully get the idea

unklerhaukus · Answer

Transitive property of equality $$\begin{cases}A = B \ A = C\end{cases}\implies\qquad  B = C$$
(consider A=B=C=1)

For non-equality$$\begin{cases}A \neq B \ A \neq C\end{cases}\qquad\not\hspace{-.5em}\implies\qquad  B \neq C$$
(consider A=1, B=2, C=2)

In other words, just because two terms are not equal equal to a third term , does not imply that the first two terms are not equal

caozeyuan · Answer

ok, but what's wrong with reflexivity?

jim_thompson5910 · Answer

$\Large x \ne x$ is false because x = x. Any number always equals itself.

caozeyuan · Answer

Got it thx

jim_thompson5910 · Answer

np