Question

For each of the following relations, determine whether it is reflexive, anti-reflexive, symmetric, anti-symmetric, or transitive. Briefly explain your answers for each one.

(a) (2 points) The domain is all CPUs. For any CPUs x and y, xRy if x has at least as many cores as y.

(b) (2 points) The domain is all people. For any people x and y, xRy if x and y are friends. Assume that everyone is his/her own friend, and that if A considers B a friend then B feels the same way about A. (This might not be true in real life, but this isn’t a social science class :)

(c) (2 points) The domain is all real numbers. For any real numbers x and y, xRy if |x–y| ≥ 2.

(d) (2 points) The domain is all real numbers. For any real numbers x and y, xRy if x+y = 0.

Answer 1

→ The objective is to determine the following relations . a) Reflective Relation Zipy if 'x' has higher clock speed than 'y'. 2. Rx in a has higher clock speed than 'n false. So No Reflexive relation. Anti - Reffective Relation :- b). to itself. je * As elementa related no * So R is Anti - Reflexive. c). Symmetrich a ry * Let a has a higher clock speed than'y'. to for symmetric yora - ie, y has beu a a higher bigher speed than 'y' (false) * It contra dice z Ry. * So, [No Symmetric Relation. d). Anti- symmetric:- * Let X. Rytaty ; then 'n has a - clock speed than 'y'. * so y hoe not higher clock speed than a

related to a). 6 . * So, * 80 y ka R is ly is not ant - Symmetric relation, e). Transistive det X. Ry & y. R2 . higher clock speed than y X. Ry a has - y. Rz y has higher clock speed 'z'. clock speed than z 1 x all has the higher above from tuanistive relation Question + 1). The domain je X. Ry if has from the above all cpu's . at least as explation: for any cous ? andy, many cores ws'y. a> 9 ya aaya and Y33 but This a> is reflexive] (symmetric and relation Transistive

2). The domain is all people. for any people x and y, XoRy if a andy are friends. Assume that Everyone is bist her own friend, and that if A considers B a friend then B feels the - same way about A. Explate nation : If a and y are friends Y 1 and a are friends then a need not be a friend of . from the above R is Ireflexive), symmetrie but not transistive. for any real numbers domin is all real 3). The numbers if (x-4132 a and y, XRy 1 2-81EO 22 la-ylə 2 lyon 72 la-yl >2 and lyral >2 12-21 22 from the I symmetric above but Relation is Jantia reflexivel not transistive .

4). The real domain numbers is x all real numbers. and y ,Ary it for many aty=0 Explanation: 2+2 0 it 200 aty=0 yta co aty=0 y+850 2+2=0 Ex: dal ; ys-1; 32) Relation is anti - reflexivel Ttranistive? symme twic] but (not) (not Anty symmetric)