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

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}

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