Could someone please check my work?

Determine which of The Three properties (reflexive, Symmetric, and Transitive) apply TO each relation. (6) Ler x be a set and let R be The relation " C" defined on subsets of X. XRX : XSX : Yes, every subser is a subset of itself, so R is Reflexive. x Ry => 4 RX : *=Y => YEX? No, R is not Symmetric. Counterexample: x = 213 9 y= El, 23 bur = El, as f x= 219 ( x Ry n y R z ) = > X RZ : ( x = Y n Y = z ) = XSZ? Yes, R is Transitive. x Suppose X EX. Then XEY by The subset rule. Likewise, Since XEY, Then X EZ. Thus, X SZ. "* likes y". ( C) Let S be The Ser of people in The School. Define R on S by x Ry iff XRX : x likes x ? No, x Might dislike itself, so R is not reflexive. XRy => yRX: x likes y implies y likes x? No, y Might dislike x, So R is ( x Ry n y Rz) = > XRz: No's likes y and y likes x, x might dislike 2, So RIS hot Transitive not symmetric