Prove that the relation R in the set of integer Z defined by R={(x,y):x-y is an integer } is an equivalence relation?
Answers!
Let x ∊ Z
(x-x)=0, which is an integer.
⇒(x,x) ∊ R
∴R is reflexive.
Let x,y ∊ Z
Let (x,y) ∊ R
⇒(x-y) is an integer
⇒-(-x y) i.e. (y-x) is also an integer
⇒(y,x) ∊ R
∴R is symmetric
Let x,y,z ∊ Z
Let (x,y),(y,z) ∊ R
⇒(x-y) and (y-z) is an integer
⇒(x-y y-z) i.e (x-z) is also an integer
⇒(x,z) ∊ R
∴R is transitive
Hence, R is an equevalence relation.
Comment
Answer the question...
Let x ∊ Z
(x-x)=0, which is an integer.
⇒(x,x) ∊ R
∴R is reflexive.
Let x,y ∊ Z
Let (x,y) ∊ R
⇒(x-y) is an integer
⇒-(-x y) i.e. (y-x) is also an integer
⇒(y,x) ∊ R
∴R is symmetric
Let x,y,z ∊ Z
Let (x,y),(y,z) ∊ R
⇒(x-y) and (y-z) is an integer
⇒(x-y y-z) i.e (x-z) is also an integer
⇒(x,z) ∊ R
∴R is transitive
Hence, R is an equevalence relation.
(x-x)=0, which is an integer.
⇒(x,x) ∊ R
∴R is reflexive.
Let x,y ∊ Z
Let (x,y) ∊ R
⇒(x-y) is an integer
⇒-(-x y) i.e. (y-x) is also an integer
⇒(y,x) ∊ R
∴R is symmetric
Let x,y,z ∊ Z
Let (x,y),(y,z) ∊ R
⇒(x-y) and (y-z) is an integer
⇒(x-y y-z) i.e (x-z) is also an integer
⇒(x,z) ∊ R
∴R is transitive
Hence, R is an equevalence relation.
Comment