Answer to Question #267864 in Discrete Mathematics for Jaishree

Question

Let R1 and R2 be the “congruent modulo 3” and the “congruent modulo 4”

relations, respectively, on the set of integers. That is, R1 = {(a, b) | a ≡ b (mod

3)} and R2 = {(a, b) | a ≡ b (mod 4)}. Find

a) R1 ∪ R2 b) R1 ∩ R2 c) R1 − R2 d) R2 − R1 e) R1 ⊕ R2

Answer 1

for R₁: a - b is divided by 3

for R₂: a - b is divided by 4

a)

$R_1 ∪ R_2=\{(a,b)|a ≡ b (mod \ 3)\ or\ a ≡ b (mod\ 4)\}$

b)

$R_1 ∩ R_2=\{(a,b)|a ≡ b (mod \ 3)\ and\ a ≡ b (mod\ 4)\}=\{(a,b)|a ≡ b (mod \ 12)\ \}$

c)

$R_1 − R_2=\{(a,b)|a ≡ b (mod \ 3)\ and\ \ not\ a ≡ b (mod\ 4)\}$

d)

$R_2 − R_1=\{(a,b)|a ≡ b (mod \ 4)\ and\ \ not\ a ≡ b (mod\ 3)\}$

e)

Symmetric Difference: R1 ⊕ R2 = {(a, b) | (a, b) ∈ R1 or (a, b) ∈ R2 but (a, b) $\notin$ R1 ∩ R2}

$R1 ⊕ R2=\{(a,b)|a ≡ b (mod \ 3)\ or\ a ≡ b (mod\ 4)\ but\ not\ a ≡ b (mod \ 12)\}$