1
GATE CSE 2007
+2
-0.6
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 %
$$10\, \le$$ b % 10 and
)a/10, b/10) $$\in \,P$$.

Consider the following ordered pairs:
$$\matrix{ {i)\,\,\,(101,\,22)} & {ii)\,\,\,(22,\,\,101)} \cr {iii)\,\,\,(145,\,\,265)} & {iv)\,\,\,(0,\,153)} \cr }$$
Which of these ordered pairs of natural numbers are comtained in P?

A
(i), (iii) and (iv)
B
(ii) and (iv)
C
(i) and (iv)
D
(iii) and (iv)
2
GATE CSE 2007
+2
-0.6
Consider the set of (column) vectors defined by $$X = \,\{ \,x\, \in \,{R^3}\,\left| {{x_1}\, + \,{x_2}\, + \,{x_3} = 0} \right.$$, where $${x^T} = \,{[{x_1}\, + \,{x_2}\, + \,{x_3}]^T}\} .$$ Which of the following is TRUE?
A
$$\left\{ {{{\left[ {1,\, - 1,\,0} \right]}^T},\,{{\left[ {1,\,\,0 ,- 1,\,} \right]}^T}} \right\}$$ is a basis for the subspace X.
B
$$\left\{ {{{\left[ {1,\, - 1,\,0} \right]}^T},\,{{\left[ {1,\,\,0,\, - 1,\,} \right]}^T}} \right\}$$ is a linearly independent set, but it does not span X and therefore is not a basis of X.
C
X is not a subspace of $${R^3}$$.
D
None of the above.
3
GATE CSE 2006
+2
-0.6
Let E, F and G be finite sets.
Let $$X = \,\left( {E\, \cap \,F\,} \right)\, - \,\left( {F\, \cap \,G\,} \right)$$
and $$Y = \,\left( {E\, - \left( {E\, \cap \,G} \right)} \right)\, - \,\left( {E\, - \,F\,} \right)$$. Which one of the following is true?
A
$$X\, \subset \,Y$$
B
$$X\, \supset \,Y$$
C
$$X\, = \,Y$$
D
$$X\, - \,Y\, \ne \,\emptyset \,\,and\,\,X\, - \,Y\, \ne \,\emptyset \,\,$$
4
GATE CSE 2006
+2
-0.6
Given a set of elements N = {1, 2, ....., n} and two arbitrary subsets $$A\, \subseteq \,N\,$$ and $$B\, \subseteq \,N\,$$, how many of the n! permutations $$\pi$$ from N to N satisfy $$\min \,\left( {\pi \,\left( A \right)} \right) = \min \,\left( {\pi \,\left( B \right)} \right)$$, where min (S) is the smallest integer in the set of integers S, and $${\pi \,\left( S \right)}$$ is the set of integers obtained by applying permutation $${\pi}$$ to each element of S?
A
$$\left( {n - \left| {A\, \cup \,B} \right|} \right)\,\left| A \right|\,\left| B \right|$$
B
$$\left( {{{\left| A \right|}^2} + {{\left| B \right|}^2}} \right)\,{n^2}$$
C
$$n!{{\left| {A\, \cap \,B} \right|} \over {\left| {A\, \cup B} \right|}}$$
D
$$\,{{{{\left| {A\, \cap \,B} \right|}^2}} \over {\left( {\matrix{ n \cr {\left| {A\, \cup \,B} \right|} \cr } } \right)}}$$
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