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Recent questions tagged proof

37 votes
5 answers
241
GATE CSE 1989 | Question: 4-i
How many substrings (of all lengths inclusive) can be formed from a character string of length $n$? Assume all characters to be distinct, prove your answer.
How many substrings (of all lengths inclusive) can be formed from a character string of length $n$? Assume all characters to be distinct, prove your answer.
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23 votes
9 answers
242
GATE CSE 1987 | Question: 10e
Show that the conclusion $(r \to q)$ follows from the premises$:p, (p \to q) \vee (p \wedge (r \to q))$
Show that the conclusion $(r \to q)$ follows from the premises$:p, (p \to q) \vee (p \wedge (r \to q))$
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15 votes
3 answers
243
GATE CSE 1987 | Question: 9c
Show that the number of odd-degree vertices in a finite graph is even.
Show that the number of odd-degree vertices in a finite graph is even.
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2 votes
0 answers
245
ISI2011-PCB-A-1
Let $D = \{d_1, d_2, \dots, d_k\}$ be the set of distinct divisors of a positive integer $n$ ($D$ includes 1 and $n$). Then show that $\Sigma_{i=1}^k \sin^{-1} \sqrt{\log_nd_i}=\frac{\pi}{4} \times k$. hint: $\sin^{−1} x + \sin^{−1} \sqrt{1-x^2} = \frac{\pi}{2}$
Let $D = \{d_1, d_2, \dots, d_k\}$ be the set of distinct divisors of a positive integer $n$ ($D$ includes 1 and $n$). Then show that$\Sigma_{i=1}^k \sin^{-1} \sqrt{\log_...
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1 votes
0 answers
246
ISI2014-PCB-A-1a
Let $x=(x_1, x_2, \dots x_n) \in \{0,1\}^n$ By $H(x)$ we mean the number of 1's in $(x_1, x_2, \dots x_n)$. Prove that $H(x) = \frac{1}{2} (n-\Sigma^n_{i=1} (-1)^{x_i})$.
Let $x=(x_1, x_2, \dots x_n) \in \{0,1\}^n$ By $H(x)$ we mean the number of 1's in $(x_1, x_2, \dots x_n)$. Prove that $H(x) = \frac{1}{2} (n-\Sigma^n_{i=1} (-1)^{x_i})$....
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21 votes
5 answers
248
GATE CSE 1991 | Question: 17,a
Show that the Turing machines, which have a read only input tape and constant size work tape, recognize precisely the class of regular languages.
Show that the Turing machines, which have a read only input tape and constant size work tape, recognize precisely the class of regular languages.
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48 votes
3 answers
249
GATE CSE 1991 | Question: 16-b
Show that all vertices in an undirected finite graph cannot have distinct degrees, if the graph has at least two vertices.
Show that all vertices in an undirected finite graph cannot have distinct degrees, if the graph has at least two vertices.
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6 votes
1 answer
250
GATE CSE 1996 | Question: 9
The Fibonacci sequence $\{f_1, f_2, f_3 \ldots f_n\}$ is defined by the following recurrence:$f_{n+2} = f_{n+1} + f_n, n \geq 1; f_2 =1:f_1=1$Prove by induction that every third element of the sequence is even.
The Fibonacci sequence $\{f_1, f_2, f_3 \ldots f_n\}$ is defined by the following recurrence:$$f_{n+2} = f_{n+1} + f_n, n \geq 1; f_2 =1:f_1=1$$Prove by induction that ev...
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19 votes
3 answers
251
GATE CSE 1995 | Question: 24
Prove that in finite graph, the number of vertices of odd degree is always even.
Prove that in finite graph, the number of vertices of odd degree is always even.
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11 votes
3 answers
252
GATE CSE 1995 | Question: 23
Prove using mathematical induction for $n \geq 5, 2^n > n^2$
Prove using mathematical induction for $n \geq 5, 2^n n^2$
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27 votes
5 answers
253
GATE CSE 1995 | Question: 21
Let $G_1$ and $G_2$ be subgroups of a group $G$. Show that $G_1 \cap G_2$ is also a subgroup of $G$. Is $G_1 \cup G_2$ always a subgroup of $G$?.
Let $G_1$ and $G_2$ be subgroups of a group $G$.Show that $G_1 \cap G_2$ is also a subgroup of $G$.Is $G_1 \cup G_2$ always a subgroup of $G$?.
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18 votes
2 answers
255
GATE CSE 1994 | Question: 15
Use the patterns given to prove that $\sum\limits_{i=0}^{n-1} (2i+1) = n^2$ (You are not permitted to employ induction) Use the result obtained in (A) to prove that $\sum\limits_{i=1}^{n} i = \frac{n(n+1)}{2}$
Use the patterns given to prove that$\sum\limits_{i=0}^{n-1} (2i+1) = n^2$(You are not permitted to employ induction)Use the result obtained in (A) to prove that $\sum\li...
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17 votes
3 answers
256
GATE CSE 1994 | Question: 5
A $3-\text{ary}$ tree is a tree in which every internal node has exactly three children. Use induction to prove that the number of leaves in a $3-\text{ary}$ tree with $n$ internal nodes is $2(n+1)$.
A $3-\text{ary}$ tree is a tree in which every internal node has exactly three children. Use induction to prove that the number of leaves in a $3-\text{ary}$ tree with $n...
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4 votes
1 answer
257
Toc theorem proof
How $\phi^*=\epsilon$?
How $\phi^*=\epsilon$?
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18 votes
3 answers
258
GATE CSE 1993 | Question: 18
Show that proposition $C$ is a logical consequence of the formula$A\wedge \left(A \to \left(B \vee C\right)\right) \wedge \left( B \to \neg A\right)$using truth tables.
Show that proposition $C$ is a logical consequence of the formula$$A\wedge \left(A \to \left(B \vee C\right)\right) \wedge \left( B \to \neg A\right)$$using truth tables....
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13 votes
3 answers
262
GATE CSE 1999 | Question: 14
Show that the formula $\left[(\sim p \vee q) \Rightarrow (q \Rightarrow p)\right]$ is not a tautology. Let $A$ be a tautology and $B$ any other formula. Prove that $(A \vee B)$ is a tautology.
Show that the formula $\left[(\sim p \vee q) \Rightarrow (q \Rightarrow p)\right]$ is not a tautology.Let $A$ be a tautology and $B$ any other formula. Prove that $(A \ve...
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34 votes
2 answers
263
GATE CSE 1999 | Question: 7
Show that the language $L = \left\{ xcx \mid x \in \left\{0,1\right\}^* \text{ and }c\text{ is a terminal symbol}\right\}$ is not context free. $c$ is not $0$ or $1$.
Show that the language $$L = \left\{ xcx \mid x \in \left\{0,1\right\}^* \text{ and }c\text{ is a terminal symbol}\right\}$$ is not context free. $c$ is not $0$ or $1$.
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40 votes
4 answers
267
GATE CSE 1992 | Question: 14a
If $G$ is a group of even order, then show that there exists an element $a≠e$, the identity in $G$, such that $a^2 = e$.
If $G$ is a group of even order, then show that there exists an element $a≠e$, the identity in $G$, such that $a^2 = e$.
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15 votes
5 answers
268
GATE1991-15,a
Show that the product of the least common multiple and the greatest common divisor of two positive integers $a$ and $b$ is $a\times b$.
Show that the product of the least common multiple and the greatest common divisor of two positive integers $a$ and $b$ is $a\times b$.
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