91影视

An equivalence relation is a relationship on a set, generally denoted by 鈥溾埣鈥�, that is reflexive, symmetric, and transitive for everything in the set.

The relation 鈥渋s equal to鈥�, denoted$=$ , is an equivalence relation on the set of real numbers since for any

1. (Reflexivity) $x=x$

2. (Symmetry) if $x=ytheny=x$

3. (Transitivity) if $x=yandy=zthenx=z$

Consider an alphabet$危.Letx鈭�{危}^{*};Fory鈭�危$

$xy鈭�Lifanonlyifxy鈭�L$

So, X is indistinguishable to X and X is arbitrary.

Therefore, $鈮�L$is reflexive

Suppose $x{鈮�}_{L}y$ . This implies that for all $z鈭�\stackrel{*}{鈭�},$

1. $xz,yz鈭�Lorxz,yz鈭�Lforallz鈭�\stackrel{*}{鈭�}.$
2. $yz,xz鈭�Loryz,xz鈭�Lforallz鈭�\stackrel{*}{鈭�}$

The implication in $\left(2\right)meansy{鈮�}_{L}x.$

Thus, $鈮�L$is symmetric.

Consider $x,y,z鈭�{危}^{*},x鈮�Ly,andy鈮�Lz.$

To prove transitivityshow that $x鈮�Lzoru鈭�{危}^{*},xu鈭�L$if and only$ifzu鈭�L.$ Consider that $u鈭�{危}^{*}andxu鈭�L.$ Now, $yu鈭�Landzu鈭�Lbecausex鈮�Lyandy鈮�Lzy$respectively. Also, $xu鈭�L.Fromx鈮�Ly,y鈮�Lz,$it is concluded that $yu鈭�Landzu鈭�L.xu鈭�L$if and only if $zu鈭�L.Itistruethatx鈮�Lz.$

Because $x,y,andz$are arbitrary, $鈮�L$is transitive.