Problem 4 (8 points) Let Z be the set of integens, and consider the relation R f

Question

Problem 4 (8 points) Let Z be the set of integens, and consider the relation R from Z to Z defined as follows R ef [a, b) e ZxZ : 42 divides a -b Which of the following statements are true and which are false? True False The relation R is reflexive. The relation R is symmetric. OO The relation R is anti-symmetric. The relation R is transitive. Now let A be an arbitrary set, and let R1. R2 be arbitrary relations from A to A. Consider the relation R1 U R2 defined as follows: Which of the following statements are true and which are false? True False o If Ri and R2 are reflexive. then R, U R2 is reflexive. OD If Ri and R2 are symmetric, then Ri U R2 is symmetric. If Rt and R2 are anti-symmetric, then R, U Rz is anti-symmetric. ODIf R1 and R2 are transitive, then R1 UR2 is transitive.

Explanation / Answer

Part1:

R is Z --> Z

(a,b) R is 42 divides a-b

1.It is reflexivs as a R a holds

42 divides a-a --> 42 divides 0

2.It is symmetric - True

If we take a = 100 b = 16

a R b holds as 42 divides 100-16 -> 42 divides 84
b R a holds as 42 divides 16-100 -> 42 divides -84

3. It is not anti symmetric as per above

   a R b and b R a does not imply a = b

4. It is transitive

   a = 100 , b = 16, c = 58

   a R b = 42 divides 84
   b R c = 42 divides -42
   a R c = 42 divides 42

   Hence proved

Part2:

R = R1 U R2

   All the statements are true because the behaviour of R is defined by the "OR" relation
   between R1 and R2. For R to satisfy any property, either R1 shoukd have that property or
   R2 should have that property. As all the questions have R1 and R2 satisfying the property,     then it is trivial to show that R should exibit that property as it is "OR" combination
   of R1 and R2.

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