GATE CSE 2007 | Set Theory & Algebra Question 42 | 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

Consider the following Hasse diagrams.
GATE CSE 2007 Discrete Mathematics - Set Theory & Algebra Question 43 English 1

GATE CSE 2007 Discrete Mathematics - Set Theory & Algebra Question 43 English 2

GATE CSE 2007 Discrete Mathematics - Set Theory & Algebra Question 43 English 3

GATE CSE 2007 Discrete Mathematics - Set Theory & Algebra Question 43 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)

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