let r be a relation over the set n times n and it is

Let r be a relation over the set $$N \times N$$ and it is defined by $$\left( {a,\,b} \right)\,r\,\left( {c,\,d} \right) \Rightarrow a + d = b + c.$$ Then $$r$$ is :

A. reflexive only

B. symmetric only

C. transitive only

D. an equivalence relation

Answer : an equivalence relation

Solution :
$$\left( {a,\,b} \right)\,r\,\left( {a,\,b} \right)$$ because $$a+b=b+a.$$
So, $$r$$ is reflexive.
$$\left( {a,\,b} \right)\,r\,\left( {c,\,d} \right) \Rightarrow a + d = b + c \Rightarrow c + b = d + a \Rightarrow \left( {c,\,d} \right)\,r\,\left( {a,\,b} \right)$$
So, $$r$$ is symmetric.
$$\left( {a,\,b} \right)\,r\,\left( {c,\,d} \right)$$ and $$\left( {c,\,d} \right)\,r\,\left( {e,\,f} \right) \Rightarrow a + d = b + c,\,\,c + f = d + e$$
Adding, $$a + d + c + f = b + c + d + e \Rightarrow a + f = b + e \Rightarrow \left( {a,\,b} \right)\,r\,\left( {e,\,f} \right)$$
$$\therefore \,r$$ is transitive.

Releted MCQ Question on
Calculus >> Sets and Relations

Releted Question 1

If $$X$$ and $$Y$$ are two sets, then $$X \cap {\left( {X \cup Y} \right)^c}$$ equals.

A. $$X$$

B. $$Y$$

C. $$\phi $$

D. None of these

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Releted Question 2

The expression $$\frac{{12}}{{3 + \sqrt 5 + 2\sqrt 2 }}$$ is equal to

A. $$1 - \sqrt 5 + \sqrt 2 + \sqrt {10} $$

B. $$1 + \sqrt 5 + \sqrt 2 - \sqrt {10} $$

C. $$1 + \sqrt 5 - \sqrt 2 + \sqrt {10} $$

D. $$1 - \sqrt 5 - \sqrt 2 + \sqrt {10} $$

View Answer

Releted Question 3

If $${x_1},{x_2},.....,{x_n}$$ are any real numbers and $$n$$ is any positive integer, then

A. $$n\sum\limits_{i = 1}^n {{x_i}^2 < {{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^2}} $$

B. $$\sum\limits_{i = 1}^n {{x_i}^2 \geqslant {{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^2}} $$

C. $$\sum\limits_{i = 1}^n {{x_i}^2 \geqslant n{{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^2}} $$

D. none of these

View Answer

Releted Question 4

Let $$S$$ = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of $$S$$ is equal to

A. 25

B. 34

C. 42

D. 41

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Let r be a relation over the set $$N \times N$$ and it is defined by $$\left( {a,\,b} \right)\,r\,\left( {c,\,d} \right) \Rightarrow a + d = b + c.$$ Then $$r$$ is :

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