Let R be the real line. Consider the following subsets of the plane R × R : S = x , y : y = x + 1 and 0 < x < 2 , T = { x , y : x - y is an integer } Which one of the following is true?

T is an equivalence relation on R but S is not.

S is an equivalence relation on R but T is not.

Neither S nor T is an equivalence relation on R .

Both S and T are equivalence relations on R .

Let W denotes the words in the English dictionary. If the relation R is given by R = { x , y ∈ W × W : the word x and y have atleast one letter in common } , then R is

reflexive, symmetric and not transitive

reflexive, symmetric and transitive

reflexive, transitive and not symmetric

symmetric, transitive and not reflexive

Check whether the relation R defined in the set 1 , 2 , 3 , 4 , 5 , 6 as R = a , b : b = a + 1 is reflexive, symmetric or transitive.

Suppose A = 1 , 2 , 3 , 4 , 5 , 6 . A relation R is defined on set A as: R = { ( a , b ) : b = a + 1 } ∴ R = { ( 1, 2 ) , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 } Reflexive Property : A relation R in a set A is said to be reflexive if x , x ∈ R , for every x ∈ A Now, 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 , 6 ∉ R ∴ R is not reflexive. Symmetric Property : A relation R in a set A is said to be symmetric if x , y ∈ R implies that y , x ∈ R for all x , y ∈ A . Now it can be observed that 1 , 2 ∈ R , but 2 , 1 ∉ R . Hence , R is not symmetric. Transitive Property : A relation R in a set A is said to be transitive if x , y ∈ R and y , z ∈ R implies that x , z ∈ R for all x , y , z ∈ A . Now, 1 , 2 , 2 , 3 ∈ R but, ( 1 , 3 ) ∉ R ∴ R is not transitive. Hence, R is neither reflexive, nor symmetric, nor transitive.