1
GATE CSE 2006
MCQ (Single Correct Answer)
+2
-0.6
Let S = {1, 2, 3,....., m} , m > 3. Let $${X_1},\,....,\,{X_n}$$ be subsets of S each of size 3. Define a function f from S to the set of natural numbers as, f (i) is the number of sets $${X_j}$$ that contain the element i. That is $$f(i) = \left\{ {j\left| i \right.\,\, \in \,{X_j}} \right\}\left| . \right.$$

Then $$\sum\limits_{i - 1}^m {f\,(i)} $$ is

A
3m
B
3n
C
2m + 1
D
2n + 1
2
GATE CSE 2006
MCQ (Single Correct Answer)
+1
-0.3
For the set $$N$$ of natural numbers and a binary operation $$f:N \times N \to N$$, an element $$z \in N$$ is called an identity for $$f$$ if $$f\left( {a,z} \right) = a = f\left( {z,a} \right)$$ for all $$a \in N$$. Which of the following binary operations have an identify?
$${\rm I}$$) $$\,\,\,\,\,\,f\left( {x,y} \right) = x + y - 3$$
$${\rm I}{\rm I}$$ $$\,\,\,\,\,\,f\left( {x,y} \right) = {\mkern 1mu} \max \left( {x,y} \right)$$
$${\rm I}{\rm I}{\rm I}$$$$\,\,\,\,\,f\left( {x,y} \right) = \,{x^y}$$
A
$${\rm I}$$ and $${\rm I}$$$${\rm I}$$ only
B
$${\rm I}$$$${\rm I}$$ and $${\rm I}$$$${\rm I}$$$${\rm I}$$ only
C
$${\rm I}$$ and $${\rm I}$$$${\rm I}$$$${\rm I}$$ only
D
None of them
3
GATE CSE 2006
MCQ (Single Correct Answer)
+1
-0.3
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
A
Neither a Partial Order nor an Equivalence Relation
B
A Partial Order but not a Total Order
C
A Total Order
D
An Equivalence relation
4
GATE CSE 2006
MCQ (Single Correct Answer)
+1
-0.3
The set $$\left\{ {1,\,\,2,\,\,3,\,\,5,\,\,7,\,\,8,\,\,9} \right\}$$ under multiplication modulo 10 is not a group. Given below are four plausible reasons.

Which one of them is false?

A
It is not closed
B
2 does not have an inverse
C
3 does not have an inverse
D
8 does not have an inverse
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