GATE CSE 2007 | Set Theory & Algebra Question 44 | Discrete Mathematics

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?

$$\left\{ {{{\left[ {1,\, - 1,\,0} \right]}^T},\,{{\left[ {1,\,\,0 ,- 1,\,} \right]}^T}} \right\}$$ is a basis for the subspace X.

$$\left\{ {{{\left[ {1,\, - 1,\,0} \right]}^T},\,{{\left[ {1,\,\,0,\, - 1,\,} \right]}^T}} \right\}$$ is a linearly independent set, but it does not span X and therefore is not a basis of X.

X is not a subspace of $${R^3}$$.

None of the above.

GATE CSE 2007

MCQ (Single Correct Answer)

-0.6

How many different non-isomorphic Abelian groups of order 4 are there?

GATE CSE 2006

MCQ (Single Correct Answer)

-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?

$$X\, \subset \,Y$$

$$X\, \supset \,Y$$

$$X\, = \,Y$$

$$X\, - \,Y\, \ne \,\emptyset \,\,and\,\,X\, - \,Y\, \ne \,\emptyset \,\,$$

GATE CSE 2006

MCQ (Single Correct Answer)

-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?

$$\left( {n - \left| {A\, \cup \,B} \right|} \right)\,\left| A \right|\,\left| B \right|$$

$$\left( {{{\left| A \right|}^2} + {{\left| B \right|}^2}} \right)\,{n^2}$$

$$n!{{\left| {A\, \cap \,B} \right|} \over {\left| {A\, \cup B} \right|}}$$

$$\,{{{{\left| {A\, \cap \,B} \right|}^2}} \over {\left( {\matrix{ n \cr {\left| {A\, \cup \,B} \right|} \cr } } \right)}}$$

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