1
GATE CSE 2004
+1
-0.3
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

A
$$\left\{ {\left( {x,y} \right)|y > x\,\,\,and\,\,\,x,y \in \left\{ {0,1,2,.....} \right\}} \right\}$$
B
$$\left\{ {\left( {x,y} \right)|y \ge x\,\,\,and\,\,\,x,y \in \left\{ {0,1,2,.....} \right\}} \right\}$$
C
$$\left\{ {\left( {x,y} \right)|y < x\,\,\,and\,\,\,x,y \in \left\{ {0,1,2,.....} \right\}} \right\}$$
D
$$\left\{ {\left( {x,y} \right)|y \le x\,\,\,and\,\,\,x,y \in \left\{ {0,1,2,.....} \right\}} \right\}$$
2
GATE CSE 2004
+1
-0.3
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 $$C$$ $$= \left\{ {1,2,3,4} \right\}$$ as defined below:

i) An element $$x$$ in $$A$$ is related to an element $$y$$ in $$B$$ (under $${R_1}$$) if $$x + y$$ is divisible by $$3$$.
ii) An element EExEE in $$B$$ is related to an elements $$y$$ in $$C$$ (under $${R_2}$$) if $$x + y$$ is even but not divisible by $$3$$.

Which is the composite relation $$R1R2$$ from $$A$$ to $$C$$?

A
$${R_1}\,{R_2}\, = \,\left\{ {\left( {1,2} \right),\,\left( {1,4} \right),\,\left( {3,3} \right),\,\left( {5,4} \right),\,\left( {7,3} \right)} \right\}$$
B
$${R_1}\,{R_2}\, = \,\left\{ {\left( {1,2} \right),\,\left( {1,3} \right),\,\left( {3,2} \right),\,\left( {5,2} \right),\,\left( {7,3} \right)} \right\}$$
C
$${R_1}\,{R_2}\, = \,\left\{ {\left( {1,2} \right),\,\left( {3,2} \right),\,\left( {3,4} \right),\,\left( {5,4} \right),\,\left( {7,2} \right)} \right\}$$
D
$${R_1}\,{R_2}\, = \,\left\{ {\left( {3,2} \right),\,\left( {3,4} \right),\,\left( {5,1} \right),\,\left( {5,3} \right),\,\left( {7,1} \right)} \right\}$$
3
GATE CSE 2001
+1
-0.3
Consider the following relations:
$${R_1}\,\,\left( {a,\,\,b} \right)\,\,\,iff\,\,\left( {a + b} \right)$$ is even over the set of integers
$${R_2}\,\,\left( {a,\,\,b} \right)\,\,\,iff\,\,\left( {a + b} \right)$$ is odd over the set of integers
$${R_3}\,\,\left( {a,\,\,b} \right)\,\,\,iff\,\,a.b > 0$$ over the set of non-zero rational numbers
$${R_4}\,\,\left( {a,\,\,b} \right)\,\,\,iff\,\,\left| {a - b} \right| \le 2$$ over the set of natural numbers

Which of the following statements is correct?

A
$${R_1}$$ and $${R_2}$$ are equivalence relations, $${R_3}$$ and $${R_4}$$ are not
B
$${R_1}$$ and $${R_3}$$ are equivalence relations, $${R_2}$$ and $${R_4}$$ are not
C
$${R_1}$$ and $${R_4}$$ are equivalence relations, $${R_2}$$ $${R_3}$$ are not
D
$${R_1}$$, $${R_2}$$, $${R_3}$$ and $${R_4}$$ are all equivalence relations
4
GATE CSE 1999
+1
-0.3
The number of binary relations on a set with $$n$$ elements is:
A
$${n^2}$$
B
$${2^n}$$
C
$$2{n^2}$$
D
None of the above
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