relations

Let F be the set of all functions from {1,..., n} to {0,1}. Define the binary relation ≤ on F as follows: ∀f,g∈F, f≤ g if and only if ∀x ∈ {1, ..., n}, f(x) ≤ g(x), where 0 ≤ 1. Wh...

A relation R is said to be circular if a R b and b R c together imply c R a. Which of the following options is/are correct?

Let $$𝑅$$ be the relation on the set of positive integers such that $$aRb$$ if and only if $$𝑎 $$ and $$𝑏$$ are distinct and have a common divisor other than $$1.$$ Which one of...

Let $$R$$ be a relation on the set of ordered pairs of positive integers such that $$\left( {\left( {p,q} \right),\left( {r,s} \right)} \right) \in R$$ if and only if $$p - s = q -...

Consider the following relation on subsets of the set S integers between 1 and 2014. For two distinct subsets U and V of S we say U < V if the minimum element in the symmetric diff...

There are two elements $$x, y$$ in a group $$\left( {G,\, * } \right)$$ such that every elements in the group can be written as a product of some number of $$x's$$ and $$y's$$ in s...

A partial order P is defined on the set of natural numbers as following. Herw x/y denotes integer division. i) (0, 0) $$ \in \,P$$. ii) (a, b) $$ \in \,P$$ if and only a % $$10\, \...

A relation $$R$$ is defined on ordered pairs of integers as follows: $$\left( {x,y} \right)R\left( {u,v} \right)\,if\,x < u$$ and $$y > v$$. Then $$R$$ is

Consider the binary relation: $$S = \left\{ {\left( {x,y} \right)|y = x + 1\,\,and\,\,x,y \in \left\{ {0,1,2,...} \right\}} \right\}$$ The reflexive transitive closure of $$S$$ is

Let $${R_1}$$ be a relation from $$A = \left\{ {1,3,5,7} \right\}$$ to $$B = \left\{ {2,4,6,8} \right\}$$ and $${R_2}$$ be another relation from $$B$$ to $$C$$ $$ = \left\{ {1,2,3,...

A relation R is defined on the set of integers as zRy if f (x + y) is even. Which of the following statements is true?

(a) Mr. X claims the following: If a relation R is both symmetric and transitive, then R is reflexive. For this, Mr. X offers the following proof. "From xRy, using symmetry we get...

Suppose $$A$$ is a finite set with $$n$$ elements. The number of elements in the Largest equivalence relation of $$A$$ is

Let R be a non-emply relation on a collection of sets defined by $${A^R}\,B $$ if and only if $$A\, \cap \,B\, = \,\phi $$. Then, (pick the true statement)

Let $$R$$ be a symmetric and transitive relation on a set $$A$$. Then

The transitive closure of the relation $$\left\{ {\left( {1,2} \right)\left( {2,3} \right)\left( {3,4} \right)\left( {5,4} \right)} \right\}$$ on the set $$A = \left\{ {1,2,3,4,5}...

(a) How many binary relations are there on a set A with n elements? (b) How many one - to - one functions are there from a set A with n elements onto itself