1
GATE CSE 1999
MCQ (Single Correct Answer)
+1
-0.3
The number of binary strings of $$n$$ zeros and $$k$$ ones such that no two ones are adjacent is:
A
$${}^{n + 1}{C_k}$$
B
$${}^n{C_k}$$
C
$${}^n{C_{k + 1}}$$
D
None of the above
2
GATE CSE 1999
Subjective
+2
-0

(a) Mr. X claims the following:
If a relation R is both symmetric and transitive, then R is reflexive. For this, Mr. X offers the following proof.

"From xRy, using symmetry we get yRx. Now because R is transitive, xRy and yRx togethrer imply xRx. Therefore, R is reflextive."


Briefly point out the flaw in Mr. X' proof.

(b) Give an example of a relation R which is symmetric and transitive but not reflexive.

3
GATE CSE 1999
MCQ (Single Correct Answer)
+2
-0.6
Two girls have picked 10 roses, 15 sunflowers and 14 daffodils. What is the number of ways they can divide the flowers among themselves?
A
1638
B
2100
C
2640
D
None of the above
4
GATE CSE 1999
Subjective
+5
-0
Let $$G$$ be a connected, undirected graph. A $$cut$$ in $$G$$ is a set of edges whose removal results in $$G$$ being broken into two or more components which are not connected with each other. The size of a cut is called its $$cardinality$$. A $$min-cut$$ of $$G$$ is a cut in $$G$$ of minimum cardinality. Consider the following graph. GATE CSE 1999 Discrete Mathematics - Graph Theory Question 30 English

(a) Which of the following sets of edges is a cut?
$$\,\,\,\,$$(i)$$\,\,\,\,\left\{ {\left( {A,\,B} \right),\left( {E,\,F} \right),\left( {B,\,D} \right),\left( {A,\,E} \right),\left( {A,\,D} \right)} \right\}$$
$$\,\,\,\,$$(ii)$$\,\,\,\,\left\{ {\left( {B,\,D} \right),\left( {C,\,F} \right),\left( {A,\,B} \right)} \right\}$$

(b) What is the cardinality of a min-cut in this graph?

(c) Prove that if a connected undirected graph $$G$$ with $$n$$ vertices has a min-cut of cardinality $$k$$, then $$G$$ has at least $$(nk/2)$$ edges.

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