GATE CSE 2007 | Set Theory & Algebra Question 41 | Discrete Mathematics

GATE CSE 2007

MCQ (Single Correct Answer)

-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?

$$\left\{ {{{\left[ {1,\, - 1,\,0} \right]}^T},\,{{\left[ {1,\,\,0 ,- 1,\,} \right]}^T}} \right\}$$ 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

Consider the following Hasse diagrams.
GATE CSE 2007 Discrete Mathematics - Set Theory & Algebra Question 41 English 1

GATE CSE 2007 Discrete Mathematics - Set Theory & Algebra Question 41 English 2

GATE CSE 2007 Discrete Mathematics - Set Theory & Algebra Question 41 English 3

GATE CSE 2007 Discrete Mathematics - Set Theory & Algebra Question 41 English 4

Which all of the above represent a lattice?

(i) and (iv) only

(ii) and (iii) only

(iii) only

(i), (ii) and (iv) only

GATE CSE 2007

MCQ (Single Correct Answer)

-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?

(i), (iii) and (iv)

(ii) and (iv)

(i) and (iv)

(iii) and (iv)

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?

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

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

$$X\, = \,Y$$

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

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