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Recent questions and answers in Discrete Mathematics

45 votes
6 answers
1
Which one of the following well-formed formulae in predicate calculus is NOT valid ? $(\forall _{x} p(x) \implies \forall _{x} q(x)) \implies (\exists _{x} \neg p(x) \vee \forall _{x} q(x))$ $(\exists x p(x) \vee \exists x q (x)) \implies \exists x (p(x) \vee q (x))$ ... $\forall x (p(x) \vee q(x)) \implies (\forall x p(x) \vee \forall x q(x))$
answered 1 day ago in Mathematical Logic Yogesh88 7.2k views
0 votes
1 answer
2
Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. the integers greater than $10$ the odd negative integers the integers with absolute value less ... $A \times Z^{+}$ where $A = \{2, 3\}$ the integers that are multiples of $10$
answered 1 day ago in Set Theory & Algebra Aman_Singh 12 views
0 votes
1 answer
3
Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. the negative integers the even integers the integers less than $100$ the real ... $\frac{1}{2}$ the positive integers less than $1,000,000,000$ the integers that are multiples of $7$
answered 1 day ago in Set Theory & Algebra Aman_Singh 14 views
0 votes
1 answer
4
Find the characteristic roots of the linear homogeneous recurrence relation $a_{n} = 2a_{n-1} - 2a_{n-2}.$ [Note: These are complex numbers.] Find the solution of the recurrence relation in part $(A)$ with $a_{0} = 1\:\text{and}\: a_{1} = 2.$
answered 1 day ago in Combinatory Aman_Singh 15 views
33 votes
4 answers
6
Let $f: A \rightarrow B$ a function, and let E and F be subsets of $A$. Consider the following statements about images. $S1: f(E \cup F) = f(E) \cup f(F)$ $S2: f(E \cap F)=f(E) \cap f(F)$ Which of the following is true about S1 and S2? Only S1 is correct Only S2 is correct Both S1 and S2 are correct None of S1 and S2 is correct
answered Jul 25 in Set Theory & Algebra Kushagra गुप्ता 4k views
5 votes
2 answers
7
Let (G,*) be a group such that O(G) = 8, where O(G) denotes the order of the group. Which of the following is True ? There exist no element a in G whose order is 6. There exist an element a in G whose order is 4. There exist more then one element in G whose order is 1 None of these
answered Jul 21 in Set Theory & Algebra Sovan Pal 408 views
23 votes
7 answers
8
Let $n =$ $p^{2}q$, where $p$ and $q$ are distinct prime numbers. How many numbers m satisfy $1 ≤ m ≤ n$ and $gcd$ $(m, n) = 1?$ Note that $gcd$ $(m, n)$ is the greatest common divisor of $m$ and $n$. $p(q - 1)$ $pq$ $\left ( p^{2}-1 \right ) (q - 1)$ $p(p - 1) (q - 1)$
answered Jul 20 in Set Theory & Algebra Jhaiyam 2.5k views
42 votes
8 answers
9
What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs $(a,b)$ and $(c,d)$ in the chosen set such that, $a \equiv c\mod 3$ and $b \equiv d \mod 5$ $4$ $6$ $16$ $24$
answered Jul 20 in Combinatory Chirag Shilwant 5.6k views
25 votes
6 answers
10
The minimum number of cards to be dealt from an arbitrarily shuffled deck of $52$ cards to guarantee that three cards are from same suit is $3$ $8$ $9$ $12$
answered Jul 20 in Combinatory Chirag Shilwant 4k views
34 votes
9 answers
11
What is the chromatic number of an $n$ vertex simple connected graph which does not contain any odd length cycle? Assume $n > 2$. $2$ $3$ $n-1$ $n$
answered Jul 19 in Graph Theory Chirag Shilwant 4.5k views
0 votes
1 answer
12
27 votes
6 answers
13
For the composition table of a cyclic group shown below: ... $a,b$ are generators $b,c$ are generators $c,d$ are generators $d,a$ are generators
answered Jul 18 in Set Theory & Algebra Kushagra गुप्ता 2.7k views
26 votes
3 answers
14
Which one of the following is false? The set of all bijective functions on a finite set forms a group under function composition. The set $\{1, 2, \dots p-1\}$ forms a group under multiplication mod $p$, where $p$ is a prime number. The set of all strings over a finite alphabet forms a group ... group $\langle G, * \rangle$ if and only if for any pair of elements $a, b \in S, a * b^{-1} \in S$.
answered Jul 18 in Set Theory & Algebra Kushagra गुप्ता 3.3k views
30 votes
11 answers
15
In how many ways can we distribute $5$ distinct balls, $B_1, B_2, \ldots, B_5$ in $5$ distinct cells, $C_1, C_2, \ldots, C_5$ such that Ball $B_i$ is not in cell $C_i$, $\forall i= 1,2,\ldots 5$ and each cell contains exactly one ball? $44$ $96$ $120$ $3125$
answered Jul 16 in Combinatory muktiseeker 4k views
3 votes
2 answers
16
Consider directed graphs on $n$ labelled vertices $\{1,2, \dots ,n\}$, where each vertex has exactly one edge coming in and exactly one edge going out. We allow self-loops. How many graphs have exactly two cycles ? $\sum_{k=1}^{n-1} k!(n-k)!$ $\frac{n!}{2}\bigg[\sum_{k=1}^{n-1} \frac{1}{k(n-k)}\bigg]$ $n!\bigg[\sum_{k=1}^{n-1} \frac{1}{k}\bigg]$ $\frac{n!(n-1)}{2}$ None of the above
answered Jul 14 in Graph Theory vaibhavkedia968 556 views
25 votes
8 answers
17
If all the edge weights of an undirected graph are positive, then any subset of edges that connects all the vertices and has minimum total weight is a Hamiltonian cycle grid hypercube tree
answered Jul 14 in Graph Theory vaibhavkedia968 2.6k views
17 votes
8 answers
18
$m$ identical balls are to be placed in $n$ distinct bags. You are given that $m \geq kn$, where $k$ is a natural number $\geq 1$. In how many ways can the balls be placed in the bags if each bag must contain at least $k$ ... $\left( \begin{array}{c} m - kn + n + k - 2 \\ n - k \end{array} \right)$
answered Jul 13 in Combinatory Jhaiyam 3.4k views
23 votes
6 answers
19
In how many ways can a given positive integer $n \geq 2$ be expressed as the sum of $2$ positive integers (which are not necessarily distinct). For example, for $n=3$, the number of ways is $2$, i.e., $1+2, 2+1$. Give only the answer ... positive integer $n \geq k$ be expressed as the sum of $k$ positive integers (which are not necessarily distinct). Give only the answer without explanation.
answered Jul 13 in Combinatory harypotter0 2k views
2 votes
1 answer
20
15. a) How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of the four aces are chosen? b) How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of the four aces and at least ... many cards must be chosen from a standard deck of 52 cards to guarantee that there are at least two cards of each of two different kinds?
asked Jul 4 in Combinatory Sanjay Sharma 163 views
3 votes
1 answer
21
How many pairs $(x,y)$ such that $x+y <= k$, where x y and k are integers and $x,y>=0, k > 0$. Solve by summation rules. Solve by combinatorial argument.
asked Jun 8 in Combinatory dd 259 views
0 votes
2 answers
24
Suppose that there are $n = 2^{k}$ teams in an elimination tournament, where there are $\frac{n}{2}$ games in the first round, with the $\frac{n}{2} = 2^{k-1}$ winners playing in the second round, and so on. Develop a recurrence relation for the number of rounds in the tournament.
asked May 10 in Combinatory Lakshman Patel RJIT 110 views
0 votes
0 answers
32
1 vote
0 answers
37
Prove Theorem $6:$Suppose that $\{a_{n}\}$ satisfies the liner nonhomogeneous recurrence relation $a_{n} = c_{1}a_{n-1} + c_{2}a_{n-2} + \dots + c_{k}a_{n-k} + F(n),$ where $c_{1}.c_{2},\dots,c_{k}$ ... is $m,$ there is a particular solution of the form $n^{m}(p_{t}n^{t} + p_{t-1}n^{t-1} + \dots + p_{1}n + p_{0})s^{n}.$
asked May 6 in Combinatory Lakshman Patel RJIT 37 views
0 votes
0 answers
38
Prove Theorem $4:$ Let $c_{1},c_{2},\dots,c_{k}$ be real numbers. Suppose that the characteristic equation $r^{k}-c_{1}r^{k-1}-\dots c_{k} = 0$ has $t$ distinct roots $r_{1},r_{2},\dots,r_{t}$ with multiplicities $m_{1},m_{2},\dots,m_{t},$ ... $\alpha_{i,j}$ are constants for $1 \leq i \leq t\:\text{and}\: 0 \leq j \leq m_{i} - 1.$
asked May 6 in Combinatory Lakshman Patel RJIT 25 views
0 votes
0 answers
39
Solve the recurrence relation $T (n) = nT^{2}(n/2)$ with initial condition $T (1) = 6$ when $n = 2^{k}$ for some integer $k.$ [Hint: Let $n = 2^{k}$ and then make the substitution $a_{k} = \log T (2^{k})$ to obtain a linear nonhomogeneous recurrence relation.]
asked May 6 in Combinatory Lakshman Patel RJIT 26 views
1 vote
0 answers
40
It can be shown that Cn, the average number of comparisons made by the quick sort algorithm (described in preamble to question $50$ in exercise $5.4),$ when sorting $n$ ... $48$ to solve the recurrence relation in part $(A)$ to find an explicit formula for $C_{n}.$
asked May 6 in Combinatory Lakshman Patel RJIT 19 views
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