GATE CSE 2005 | Set Theory & Algebra Question 59 | Discrete Mathematics

GATE CSE 2005

MCQ (Single Correct Answer)

-0.3

The set $$\left\{ {1,\,\,2,\,\,4,\,\,7,\,\,8,\,\,11,\,\,13,\,\,14} \right\}$$ is a group under multiplication modulo $$15$$. The inverse of $$4$$ and $$7$$ are respectively:

$$3$$ and $$13$$

$$2$$ and $$11$$

$$4$$ and $$13$$

$$8$$ and $$14$$

GATE CSE 2004

MCQ (Single Correct Answer)

-0.3

The number of different $$n$$ $$x$$ $$n$$ symmetric matrices with each elements being either $$0$$ or $$1$$ is (Note: power ($$2,$$ $$x$$) is same as $${2^x}$$)

power $$(2, n)$$

power $$\left( {2,\,{n^2}} \right)$$

$$\left( {2,\left( {{n^2} + n} \right)/2} \right)$$

power $$\left( {2,\left( {{n^2} - n} \right)/2} \right)$$

GATE CSE 2004

MCQ (Single Correct Answer)

-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

$$\left\{ {\left( {x,y} \right)|y > x\,\,\,and\,\,\,x,y \in \left\{ {0,1,2,.....} \right\}} \right\}$$

$$\left\{ {\left( {x,y} \right)|y \ge x\,\,\,and\,\,\,x,y \in \left\{ {0,1,2,.....} \right\}} \right\}$$

$$\left\{ {\left( {x,y} \right)|y < x\,\,\,and\,\,\,x,y \in \left\{ {0,1,2,.....} \right\}} \right\}$$

$$\left\{ {\left( {x,y} \right)|y \le x\,\,\,and\,\,\,x,y \in \left\{ {0,1,2,.....} \right\}} \right\}$$

GATE CSE 2004

MCQ (Single Correct Answer)

-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 $$C$$?

$${R_1}\,{R_2}\, = \,\left\{ {\left( {1,2} \right),\,\left( {1,4} \right),\,\left( {3,3} \right),\,\left( {5,4} \right),\,\left( {7,3} \right)} \right\}$$

$${R_1}\,{R_2}\, = \,\left\{ {\left( {1,2} \right),\,\left( {1,3} \right),\,\left( {3,2} \right),\,\left( {5,2} \right),\,\left( {7,3} \right)} \right\}$$

$${R_1}\,{R_2}\, = \,\left\{ {\left( {1,2} \right),\,\left( {3,2} \right),\,\left( {3,4} \right),\,\left( {5,4} \right),\,\left( {7,2} \right)} \right\}$$

$${R_1}\,{R_2}\, = \,\left\{ {\left( {3,2} \right),\,\left( {3,4} \right),\,\left( {5,1} \right),\,\left( {5,3} \right),\,\left( {7,1} \right)} \right\}$$

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