Short Answer TypeShow that the relation R in the set A of all the books in a library of a college given by R = {(x, y): x and y have same number of pages} is an equivalence relation.
A is the set of all books in a library of a college.
R = {(x,y) : x and y have same number of pages}
Since (x, x) ∈ R as x and x have the same number of pages ∀ x ∈ A.
∴ R is reflexive.
Also (x, y) ∈ R
⇒ x and y have the same number of pages ⇒ y and x have the same number of pages
⇒ (y, x) ∈ R
∴ R is symmetric.
Now, (x, y) ∈ R and (y, z) ∈ R.
⇒ x and y have the same number of pages and y and z have the same number of pages
⇒ x and z have he same number of pages ⇒ (x, z) ∈ R ∶ R is transitive.
Determine whether each of the following relations are reflexive, symmetric and transitive :
(i) Relation R in the set A = {1, 2, 3,....., 13, 14} defined as R = {(x, y) : 3 x – y = 0}
Determine whether each of the following relations are reflexive, symmetric and transitive :
(ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}
Determine whether each of the following relations are reflexive, symmetric and transitive :
(ii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x,y) : y is divisible by x}
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