Answer to question on consumer choice

In general, if C({x,y})={x}, then x \succsim y & not「y \succsim x」 so that x \succ y holds. ...(*)

One student asked me that it seems contradicting to the statement in
the first example of p.6 (Section 2.3) in resume:
\succsim which rationalizes C(.) s.t. C({x,y})={x}, C({x,y,z})=x
is given by ① x \succsim y, x \succsim z, y \succsim z.

Using (*), you might think that it should be written as
"\succsim which satisfies 「 ①' x \succ y, x \succ z, y \succsim z」 rationalizes C(.). "

Here is Prof. Kamiya's reply:
In general, binary relation is represented using only \succsim
(and not \succ, \sim ...etc). Since ① does NOT contain
"y \succsim x", it implies that "y \succsim x" does NOT hold. Hence we can conclude that
「x \succsim y & not「y \succsim x」」 so that x \succ y.
Thus, representating \succsim by ① is correct.