3. (12 pts) Determine whether the following binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric,...

bra then Ⓡi) Reflexiven ara 5a-a {An integer a can be equal to itself. 3 so, ara is true. Hence it is reflexive (; if a Rb then > if a=b then bza. Hence, it is symmetric (iii) If arb and brc arc = ? > a=b and b=c asci a (as a, b, c E 2 hence it is true). So, it is transitive (iv) arb , bra and a ab It is autrymmetic Set of people arb as a>b.

(a is taller than b). a Ra a should be taller than a. Jot means a person should be taller than himself which is possible. So it is not reflexive. if arb then bra.

& if ais taller than then b should be taller thana. But it is not true either a ar b will be taller than other. So, If arb then bra is not possible. Hence it is not symmetric. b ).

Not

If arba brc then arc. s if a's taller than b and bistaller than than a should be taller than cehich is fue. a if arb and brc a QRC. Est is transitive (Rus arb and bra then a=b. But here arb doenot reply bra.

Because one of them can be talle from other. If a=6 it means they have equal height s, it is not antisymmetrica (a,b,c,d)- is assumed tears to define relation. a b C ③ (+) MR = a O C Rss (0,0); (a,b); (b,9); (6,6), (b,c); (5,5); should be [ca,d) } (" for this relations to be reflexive all diagonal element 1 is R should have (9,a); (b, b), (c, c) elid) But (cc) is missing. Hence it is not refleire. 9 for symmetric relation aij = aji = 4.

Here, apart from has lies oདད , 022 2 quỵ , 9,* 92 , 9234932 for arb a bra. Hence it is symmetric

then it is symmetric M- 0 6 1110 - - ооо1 6 0 ol Also if Mant AMT= IM=MT (Hence, it is Symmetric. ) (c) To check for transitivity. if aij =) -1 and Ojk = 1 = like = 1 it means if i Rj, j R k then iRk. But here in relation, a,2=1, azzal but a, 3t 1.

So, this is not transitive Idy for autisymmetric :- → It allow diagonal elements. But not the handiagonal elements. In our case - are not R-{ca, a); (a,b); Cb, a); (b,b), Cb,c); cc,b); (d,d)3 Bet here along with 19,9); (b,b), (e, c) we have (b, a)-> (415) 3 they together (ab) (bico permissible. had aly (ba), (b, c) alaugurth ajagal elements it would have been autisymmetric ; But here (1,5) Cas b) are also foresent: So, it is not antrisymmetrical If we had