Show that the relation R defined by R={(a,b):a-b is divisible by 3;a,b E N}is an equivalence relation - Maths - Relations and Functions

Question

Show that the relation R defined by R={(a,b):a-b is divisible
by 3;a,b E N}is an equivalence relation

ramandeep.singh · Accepted Answer

Reflexivivity: For any $a \in N$ , we have $a - a = 0 = 0 \times 3 \Rightarrow a - a i s d i v i s i b l e b y 3 \Rightarrow (a, a) \in R$ Thus, $(a, a) \in R$ for all $a \in N$ So, R is reflexive Symmetricity: Let $(a, b) \in R$ $\Rightarrow (a - b) i s d i v i s i b l e b y 3 \Rightarrow (a - b) = 3 p f o r s o m e p \in N \Rightarrow (b - a) = 3 (- p) \Rightarrow (b - a) i s d i v i s i b l e b y 3 \Rightarrow (b, a) \in R$ Thus, $(a, b) \in R$ $\Rightarrow (b, a) \in R$ for all $a, b \in N$ So, R is symmetric Transitivity:Let $a, b, c \in N$ such that $(a, b) \in R$ and $(b, c) \in R$ Then, $(a, b) \in R \Rightarrow (a - b) i s d i v i s i b l e b y 3 \Rightarrow (a - b) = 3 p f o r s o m e p \in N (b, c) \in R \Rightarrow (b - c) i s d i v i s i b l e b y 3 \Rightarrow (b - c) = 3 q f o r s o m e q \in N ∴ (a, b) \in R a n d (b, c) \in R \Rightarrow (a - b) = 3 p a n d (b - c) = 3 q \Rightarrow (a - b) + (b - c) = 3 p + 3 q \Rightarrow (a - c) = 3 (p + q) \Rightarrow (a - c) i s d i v i s i b l e b y 3 \Rightarrow (a, c) \in R$ So, R is transitive Thus, R is an equivalence relation

Show that the relation R defined by R={(a,b):a-b is divisible by 3;a,b E N}is an equivalence relation .