Question

Math 240 Assignment 4 - due Friday, February 28 each relation R defined on the given set A, determine whether or not it is reflexive, symmetric, anti-symmetric, or transitive. Explain why. (a) A = {0, 1,2,3), R = {(0,0).(0,1),(1,1),(1,2).(2, 2), (2.3)} (b) A = {0, 1,2,3), R = {(0,0).(0,2), (1,1),(1,3), (2,0), (2,2), (3,1),(3,3)} (c) A is the set of all English words. For words a and b, (a,b) E R if and only if a and b have at least one letter in common. (d) A is the set of all English words. For words a and b, (a,b) E R if and only if a and b are the same length (e) A = Z, (a,b) e Riff ab > 0. (f) A is the set of all logical statements. For statements p and g, (p, q) E R if and only if p +q is true. (g) A = Z, (a, b) e R if and only if (al = 161. (h) A = Z, (a,b) E R if and only if a - b is divisible by 2. 2. Which of the relations in Problem (1) are equivalence relations? For each relation that is an equivalence relation, describe the set of all equivalence classes.

Answer 1

1. COM A = {0, 1, 2, 3 } R = f(0,0), (01), (1, 1), (1, 2), (2, 2), (2, 3) } is It is not reflexive as (3,3) & R. (id It is not symmetric as a) ER but ing It is not transitive as (2) and (2, 3) eR civo It is anti-symmetric. (10) ¢ R. but (1,3)&R b) R = $(0,0), (0,2), (),1), (1,3), (2,0), (1,2), (3, 12, (3, 3)} ů (0,0), (1, 1), (8,2), (3,3) EX (Reflenitve) (0,0), (1, 1), (2, 2), (3, 3) ER symmetric) (0, 2), (2,0) ER (3, 1), (1,3) GR ta,b), (6, c) ER, (A,C)ER (Transitive). not anti - Symmetric. C For a and b, (a,b) E Rith a and to have atleast one letter common. - Reflexive clearly, Ca, agER (as a and a have all letters common If (a, b) eR (6, a) ER (Symmetric) it a and b have at least one letter common then banda " " " " ". Not Transitive as Cone, three) ER (letter 'e' is common (three, Hit) er letter 'h' is common but (one , Hit) & R as not letter is in common to both the words one and hit . Hence it is Reflerive, symmetric, not Transitive and not anti-symmetric.

i us for any de us for any english word a, we have ca, as op (inin a and a has sume length Reytenie in It (a, b) o R then Lager ie, 4 a and to have some length the same length (symmernes then hand a alen have in If (a,b), (b, c) eR, then and there fure (a, c) eR. a anda have some (Troms thre) length er not anti-symmetric as it is synmorse. ( Ac TL, Co, 2) ск, it o, 20 . clearly tact, a, a) er because product of two positive and two negative numbers are again positive) and it aso, then a a=ozo So, R is reflective. (a, b) GR, then ab eo bazo (since alaba) commutative low (ba) eR (Symmetric) a suppose ng Not Transitive as (-1,0) GR but (-1, 1) Q R and (0, 1) eR as f1 (1) = -120. is not anti-symmetric as it is symmetric.

classmate Date Page (t) (hq) ER itt p q is true cij Reflenite as pyp is always true. (1) Not Symmetric as P q is true need not implies thing eg. - If I hit my thumb with a hammer, then my thumb will hurt Here, p = f I hit my thumb with a hammer. - q = my thumb will hurt .. Here p q but q top. in Transitive as p q and q r implies P a . (g) (a, b) ER iff 1a/= 161 I c clearly, a aj ER u ff (a, b) ER, then fatz as Tal= 1a (Reflexive) 104=16) implies (b, a) er ta bez (Symmetric )! cing It (a, b), (b, c) ER then lal=16) 2 = lal= 1cl. 161 = 10 (a, C) ER ( Transitive) and (2,-2) ER anti-symmetric as (-2, 2) E R as 1-2)=121 but 2 7-2. (iv) Not

ceber q abis dirisible by 2. o clearly, a ager as osa-a is always civisible by 2. Suproke 6, b) ER se, a-b is durisible by 2. then, B. b) is also divisible by 2 1 bath Caymmerie) 1 1 1 1 1 I a.) ER tren i e, arb is divible by 2 at = 28 for some rez 4 1 1 b, c) ER Now, - then b-c=272 for some rez a-c = 9(%, +52) = 262, 8z EU = 2 Y- 1 1 Pence, a cjer (Transitive). 1 1 1 but 28-2 as NT ant symmetric (5,-2) ER -3.91 ER 1 1 (6 (0) 1 and 15 relation. are equivalence 1 5 Ledef921, D 1,3), (23:40,23, 131, 134 مش کل را دور حل کے بزنیم بیت الر ر وائی the word with length I will come J = 5 this set all Esmirajence Janesis of DIR. [2] 1,024 t uce Costis 12.1 s Set oewivalence comes is 1022,34-4,353,-.