1
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
$$X\, - \,Y\, \ne \,\emptyset \,\,and\,\,X\, - \,Y\, \ne \,\emptyset \,\,$$
2
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
$$\,{{{{\left| {A\, \cap \,B} \right|}^2}} \over {\left( {\matrix{ n \cr {\left| {A\, \cup \,B} \right|} \cr } } \right)}}$$
3
GATE CSE 2006
+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
4
GATE CSE 2006
+2
-0.6
Consider the set S = {a, b, c, d}. Consider the following 4 partitions $$\,{\pi _1},\,{\pi _2},\,{\pi _3},\,{\pi _4}$$ on $$S:\,{\pi _1} = \left\{ {\overline {a\,b\,c\,d} } \right\},\,{\pi _2} = \left\{ {\overline {a\,b\,} ,\,\overline {c\,d} } \right\},\,{\pi _3} = \left\{ {\overline {a\,b\,c\,} ,\,\overline d } \right\},\,{\pi _4} = \left\{ {\overline {a\,} ,\,\overline b ,\,\overline c ,\,\overline d } \right\}.$$ Let $$\prec$$ be the partial order on the set of partitions $$S' = \{ {\pi _1},\,{\pi _2},\,{\pi _3},\,{\pi _4}\}$$ defined as follows: $${\pi _i} \prec \,\,{\pi _j}$$ if and only if $${\pi _i}$$ refines $${\pi _j}$$. The poset diagram for $$(S',\, \prec )$$ is
A
B
C
D
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