Question

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3. Relations - No Proofs! Determine (no proof needed!) whether each of the following relations R, S, T on the set of real numbers is reflexive, symmetric, antisymmet- ric, and/or transitive. a) x Ry iff 3 - y is positive: reflexive: symmetric: anti-symmetric: transitive: b) xSy iff 2 = 2y: reflexive: symmetric anti-symmetric: transitive: c) Ty iff zy 30: reflexive: symmetric: anti-symmetric transitive:

Answer 1

x-x=0 = anya, yer a) x Ry iff x-y is positive but o is not positive XRx ;. ex not relates a - R is not reflexive Now assume a Ry =) x-y is positive y-x=-(a-y) =) yox is negative = y doesn't relatex - a Ry XRy but y Rx for - R is not symmetric relation Now, Assume XRy = x x-y is positive 4 yox is positive But this is not possible Since if x-y is positive then y-x = -(x-y) which is negative the given relation is not antisymmetric fyra

Now, assume XRy, y Rz a-y & y-z are positive but x-z = (x-7)+(y-2) =) Sum of two positive seal numbers positive real aumber x-z is positive a Rz R is transitive IS given relation 2 EIR then 2 c asy iff x=2y consider + 2(2) i-e 2 #4 282 2 doesn't relate 2 :) I is not reflexive Assume ary = x=24

= y = 0 x = = 80 (y + 2x, y = 2 s is not symmetric Now assume asy & ysa = 2y fy y = 2x x=2(2x) x= 4X = a-4250 X (1-4)=0 Certy almost og is not antisymmetric Now, assume xgy -). 2=2 x = 24 y = 22 = 2(22) - x = 42 ie x 722 * &Z = S is not transitive a=0 fysz f

2 C) a Ty iff ayco Now, x = x x 70 =) xta i ex doesn't relate a = I is not reflexive Now, assume aty =) xyco =) youso Cay=yx) =) y Tx I is symmetric relation Now, assume aty ਤ Consider (2) (-3) (-3) T2 ie 2 (-3) so (-3). (2) < 0 - But 27 (3) I is not antisymmetric Now, assuppe try & gr 2 TED & -T3 2(-1) = (2) <o & (-1) 3) = (3 f E +13) But 2.3=670

273 = T is not transitive relation