Hi! I\'m having some trouble with this DISCRETE MATH problem! Can anyone please

Question

Hi! I'm having some trouble with this DISCRETE MATH problem! Can anyone please help?

Relations:

i. Let R be a binary relation defined on the set of all integers Z as follows:

for all integers m and n

a. Is R reflective?

b. Is R symmetric?

c. Is R transitive?

d. Is R an equivalence relation?

ii. Let R1 = {(1,2),(2,3),(3,4)} and R2 = {(1,1),(1,2),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(3,4)} be relations from {1,2,3} to {1,2,3,4}. Find:

c. R1 - R2

d. R2 - R1

f. Represent R1 and R2 using matrix

g. [1]R2

h. R12   

Any help will be greatly appreciated! Thanks!

Explanation / Answer

1)

i) Let m -> Z. Then m-m is divisible by 5. Therefore mRMa holds for all a in Z and R is reflexive.

(ii) Let m,n -> Z and aRb hold. Then m-n is divisible by 5 and therefore n-m is divisible by 5.

Thus, mRn ? nRm and therefore R is symmetric.

(iii) Let m,n,k ? Z and mRn, nRm both hold. Then m-n and n-k are both divisible by 5.

Therefore m-k = (m-n) + (n-k) is divisible by 5.

Thus, mRn and nRk -> mRk and therefore R is transitive.

Since R is reflexive, symmetric and transitive so, R is an equivalence relation on Z.

2) a ) { ( 1 , 2 ) , ( 2 , 3 ) , ( 3 , 4 ) , ( 1 , 1 ) , ( 2 , 1 ) , ( 2 , 2 ) , ( 3 , 1 ) , ( 3 , 2 ) , ( 3 , 3 ) }
b ) R 1 n R 2 = {( 1 , 2 ) , ( 2 , 3 ) , ( 3 , 4 ) }
c ) R 1 – R 2 = { Ø }
d ) R 2 – R 1 = { ( 1 , 1 ) , ( 2 , 1 ) , ( 2 , 2 ) , ( 3 , 1 ) , ( 3 , 2 ) , ( 3 ,3 ) }

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