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Recent questions and answers in Discrete Mathematics

0 votes
3 answers
1
Let $G=(V, E)$ be an undirected unweighted connected graph. The diameter of $G$ is defined as: diam$(G)=\displaystyle \max_{u,v\in V} \{$the length of shortest path between $u$ and $v\}$ Let $M$ be the adjacency matrix of $G$. Define graph $G_2$ on the same set of vertices with adjacency ... diam$(G)/2\rceil<$diam$(G_2)<$ diam$(G)$ diam$(G_2)$ = diam$(G)$ diam$(G)<$ diam$(G_2)\leq 2 $ diam$(G)$
answered 12 hours ago in Graph Theory ascend 196 views
20 votes
5 answers
2
Choose the correct alternatives (More than one may be correct). Indicate which of the following well-formed formulae are valid: $\left(P\Rightarrow Q\right) {\wedge} \left(Q \Rightarrow R\right) \Rightarrow \left(P \Rightarrow R\right)$ ...
answered 1 day ago in Mathematical Logic subbus 3.6k views
2 votes
4 answers
3
A relation $R$ is said to be circular if $aRb$ and $bRc$ together imply $cRa$. Which of the following options is/are correct? If a relation $S$ is reflexive and symmetric, then $S$ is an equivalence relation. If a relation $S$ is circular and symmetric, ... and circular, then $S$ is an equivalence relation. If a relation $S$ is transitive and circular, then $S$ is an equivalence relation.
answered 1 day ago in Set Theory & Algebra Vallabh Mandare 329 views
10 votes
7 answers
4
Answer the following: Which of the following well-formed formulas are equivalent? $P \rightarrow Q$ $\neg Q \rightarrow \neg P$ $\neg P \vee Q$ $\neg Q \rightarrow P$
answered 1 day ago in Mathematical Logic subbus 1.4k views
13 votes
6 answers
5
Show that the conclusion $(r \to q)$ follows from the premises: $p, (p \to q) \vee (p \wedge (r \to q))$
answered 2 days ago in Mathematical Logic subbus 1.4k views
0 votes
4 answers
6
Let $p$ and $q$ be two propositions. Consider the following two formulae in propositional logic. $S_1: (\neg p\wedge(p\vee q))\rightarrow q$ $S_2: q\rightarrow(\neg p\wedge(p\vee q))$ Which one of the following choices is correct? Both $S_1$ and $S_2$ are tautologies ... tautology but $S_2$ is not a tautology $S_1$ is not a tautology but $S_2$ is a tautology Niether $S_1$ nor $S_2$ is a tautology
answered 2 days ago in Mathematical Logic Harshq 177 views
0 votes
2 answers
7
Given a grid of $4\times4$ points,how many triangles with their vertices on the grid can be drawn?
answered 5 days ago in Combinatory reboot 346 views
21 votes
3 answers
8
What is the possible number of reflexive relations on a set of 5 elements? $2^{10}$ $2^{15}$ $2^{20}$ $2^{25}$
answered 5 days ago in Set Theory & Algebra kshitij86 3.8k views
5 votes
4 answers
9
Match List-I with List-II: ... - (iv) (a) - (iv); (b) - (i); (c) - (iii); (d) - (ii) (a) - (iv); (b) - (iii); (c) - (i); (d) - (ii)
answered Feb 20 in Mathematical Logic anjli 763 views
0 votes
2 answers
10
In propositional logic if $\left ( P \rightarrow Q \right )\wedge \left ( R \rightarrow S \right )$ and $\left ( P \vee R \right )$ are two premises such that $\begin{array}{c} (P \to Q) \wedge (R \to S) \\ P \vee R \\ \hline Y \\ \hline \end{array}$ $Y$ is the premise : $P \vee R$ $P \vee S$ $Q \vee R$ $Q \vee S$
answered Feb 20 in Mathematical Logic anjli 328 views
0 votes
3 answers
12
1 vote
3 answers
13
Which of the following statements is false? $(P\land Q)\lor(\sim P\land Q)\lor(P \land \sim Q)$ is equal to $\sim Q\land \sim P$ $(P\land Q)\lor(\sim P\land Q)\lor(P \wedge \sim Q)$ is equal to $Q\lor P$ $(P\wedge Q)\lor (\sim P\land Q)\lor(P \wedge \sim Q)$ is equal to $Q\lor (P\wedge \sim Q)$ $(P\land Q)\lor(\sim P\land Q)\lor (P \land \sim Q)$ is equal to $P\lor (Q\land \sim P)$
answered Feb 20 in Mathematical Logic anjli 180 views
0 votes
4 answers
14
0 votes
2 answers
15
Which of the following is FALSE? $Read\ \wedge as\ AND, \vee\ as\ OR, \sim as\ NOT, \rightarrow$ as one way implication and $\leftrightarrow$ as two way implication? $((x\rightarrow y)\wedge x)\rightarrow y$ $((\sim x\rightarrow y)\wedge (\sim x\wedge \sim y))\rightarrow x$ $(x\rightarrow (x\vee y))$ $((x\vee y)\leftrightarrow (\sim x\vee \sim y))$
answered Feb 20 in Mathematical Logic anjli 127 views
0 votes
2 answers
16
38 votes
12 answers
17
1 vote
2 answers
18
Which of the following pairs of propositions are not logically equivalent? $((p \rightarrow r) \wedge (q \rightarrow r))$ and $((p \vee q) \rightarrow r)$ $p \leftrightarrow q$ and $(\neg p \leftrightarrow \neg q)$ $((p \wedge q) \vee (\neg p \wedge \neg q))$ and $p \leftrightarrow q$ $((p \wedge q) \rightarrow r)$ and $((p \rightarrow r) \wedge (q \rightarrow r))$
answered Feb 19 in Discrete Mathematics anjli 235 views
3 votes
3 answers
19
How many partial functions are there from a set with m elements to a set with n elements? Q. I cannot get the intuition how the solution arrived to be (n+1)^m
answered Feb 19 in Set Theory & Algebra 5a1n1amarjeet 955 views
2 votes
1 answer
20
There are $6$ jobs with distinct difficulty levels, and $3$ computers with distinct processing speeds. Each job is assigned to a computer such that: The fastest computer gets the toughest job and the slowest computer gets the easiest job. Every computer gets at least one job. The number of ways in which this can be done is ___________.
answered Feb 19 in Combinatory Arjun 466 views
1 vote
2 answers
21
Let $S$ be a set of consisting of $10$ elements. The number of tuples of the form $(A,B)$ such that $A$ and $B$ are subsets of $S$, and $A \subseteq B$ is ___________
answered Feb 19 in Set Theory & Algebra Shaik Masthan 358 views
1 vote
2 answers
22
Choose the correct choice(s) regarding the following proportional logic assertion $S$: $S: (( P \wedge Q) \rightarrow R) \rightarrow (( P \wedge Q) \rightarrow (Q \rightarrow R))$ $S$ is neither a tautology nor a contradiction $S$ is a tautology $S$ is a contradiction The antecedent of $S$ is logically equivalent to the consequent of $S$
answered Feb 19 in Mathematical Logic Shaik Masthan 335 views
2 votes
2 answers
23
Let $G$ be a group of order $6$, and $H$ be a subgroup of $G$ such that $1<|H|<6$. Which one of the following options is correct? Both $G$ and $H$ are always cyclic $G$ may not be cyclic, but $H$ is always cyclic $G$ is always cyclic, but $H$ may not be cyclic Both $G$ and $H$ may not be cyclic
answered Feb 18 in Set Theory & Algebra Sanjay Sharma 198 views
1 vote
1 answer
24
For two $n$-dimensional real vectors $P$ and $Q$, the operation $s(P,Q)$ is defined as follows: $s(P,Q) = \displaystyle \sum_{i=1}^n (P[i] \cdot Q[i])$ Let $\mathcal{L}$ be a set of $10$-dimensional non-zero real vectors such that for every pair of distinct vectors $P,Q \in \mathcal{L}$, $s(P,Q)=0$. What is the maximum cardinality possible for the set $\mathcal{L}$? $9$ $10$ $11$ $100$
answered Feb 18 in Set Theory & Algebra zxy123 212 views
1 vote
1 answer
25
Consider the following sets, where $n \geq 2$: $S_1$: Set of all $n \times n$ matrices with entries from the set $\{ a, b, c\}$ $S_2$: Set of all functions from the set $\{0,1,2, \dots, n^2-1\}$ to the set $\{0, 1, 2\}$ Which of the following ... to $S_2$ There exists a surjection from $S_1$ to $S_2$ There exists a bijection from $S_1$ to $S_2$ There does not exist an injection from $S_1$ to $S_2$
answered Feb 18 in Set Theory & Algebra zxy123 320 views
0 votes
2 answers
26
The number of edges in a complete graph with $‘n’$ vertices is equal to : $n(n-1)$ $\large\frac{n(n-1)}{2}$ $n^2$ $2n-1$
answered Feb 16 in Graph Theory Hira Thakur 160 views
2 votes
1 answer
27
Let G(V,E) be a simple graph. Let G’(V,E’) be a graph obtained from G such that (u,v) is an edge in G’ if (u,v) is not an edge in G. Which of the following is true? At least one of G or G’ are connected. G is necessarily disconnected. Both G and G’ are disconnected. None of the above.
answered Feb 16 in Graph Theory Agrek11 99 views
1 vote
0 answers
28
For two n-bit strings x, y ∈ {0, 1}n, define z := x ⊕ y to be the bitwise XOR of the two strings (that is, if xi, yi, zi denote the i-th bits of x, y, z respectively, then zi = xi + yi mod 2). A function h : {0, 1}n → {0, 1}n is called linear if h(x ⊕ y) = h(x) ⊕ h(y), for every x, y ∈ {0, 1}n. The number of such linear functions for n ≥ 2: 2^n 2^2n 2^(n+1) n
asked Feb 15 in Combinatory vivek_mishra 31 views
1 vote
3 answers
29
The number of positive integers not exceeding $100$ that are either odd or the square of an integer is _______ $63$ $59$ $55$ $50$
asked Nov 20, 2020 in Set Theory & Algebra jothee 372 views
2 votes
2 answers
30
How many ways are there to pack six copies of the same book into four identical boxes, where a box can contain as many as six books? $4$ $6$ $7$ $9$
asked Nov 20, 2020 in Combinatory jothee 271 views
0 votes
1 answer
31
Let $G$ be a directed graph whose vertex set is the set of numbers from $1$ to $100$. There is an edge from a vertex $i$ to a vertex $j$ if and only if either $j=i+1$ or $j=3i$. The minimum number of edges in a path in $G$ from vertex $1$ to vertex $100$ is ______ $23$ $99$ $4$ $7$
asked Nov 20, 2020 in Discrete Mathematics jothee 131 views
0 votes
1 answer
32
If $f(x)=x$ is my friend, and $p(x) =x$ is perfect, then correct logical translation of the statement “some of my friends are not perfect” is ______ $\forall _x (f(x) \wedge \neg p(x))$ $\exists _x (f(x) \wedge \neg p(x))$ $\neg (f(x) \wedge \neg p(x))$ $\exists _x (\neg f(x) \wedge \neg p(x))$
asked Nov 20, 2020 in Discrete Mathematics jothee 95 views
0 votes
1 answer
33
What kind of clauses are available in conjunctive normal form? Disjunction of literals Disjunction of variables Conjunction of literals Conjunction of variables
asked Nov 20, 2020 in Discrete Mathematics jothee 66 views
0 votes
1 answer
34
Consider the following properties: Reflexive Antisymmetric Symmetric Let $A=\{a,b,c,d,e,f,g\}$ and $R=\{(a,a), (b,b), (c,d), (c,g), (d,g), (e,e), (f,f), (g,g)\}$ be a relation on $A$. Which of the following property (properties) is (are) satisfied by the relation $R$? Only $a$ Only $c$ Both $a$ and $b$ $b$ and not $a$
asked Nov 20, 2020 in Discrete Mathematics jothee 115 views
0 votes
1 answer
35
Consider the following argument with premise $\forall _x (P(x) \vee Q(x))$ and conclusion $(\forall _x P(x)) \wedge (\forall _x Q(x))$ ... valid argument Steps $(C)$ and $(E)$ are not correct inferences Steps $(D)$ and $(F)$ are not correct inferences Step $(G)$ is not a correct inference
asked Nov 20, 2020 in Discrete Mathematics jothee 57 views
0 votes
1 answer
36
Consider the following statements: Any tree is $2$-colorable A graph $G$ has no cycles of even length if it is bipartite A graph $G$ is $2$-colorable if is bipartite A graph $G$ can be colored with $d+1$ colors if $d$ is the maximum degree of any vertex in the graph $G$ ... and $(e)$ are incorrect $(b)$ and $(c)$ are incorrect $(b)$ and $(e)$ are incorrect $(a)$ and $(d)$ are incorrect
asked Nov 20, 2020 in Discrete Mathematics jothee 103 views
0 votes
1 answer
37
Consider the statement below. A person who is radical $(R)$ is electable $(E)$ if he/she is conservative $(C)$, but otherwise not electable. Few probable logical assertions of the above sentence are given below. $(R \wedge E) \Leftrightarrow C$ $R \rightarrow (E \leftrightarrow C)$ ... answer from the options given below: $(B)$ only $(C)$ only $(A)$ and $(C)$ only $(B)$ and $(D)$ only
asked Nov 20, 2020 in Discrete Mathematics jothee 73 views
0 votes
0 answers
38
Let $G$ be a simple undirected graph, $T_D$ be a DFS tree on $G$, and $T_B$ be the BFS tree on $G$. Consider the following statements. Statement $I$: No edge of $G$ is a cross with respect to $T_D$ Statement $II$: For every edge $(u,v)$ of $G$ ... Statement $I$ and Statement $II$ are false Statement $I$ is correct but Statement $II$ is false Statement $I$ is incorrect but Statement $II$ is true
asked Nov 20, 2020 in Discrete Mathematics jothee 61 views
2 votes
1 answer
39
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, 2020 in Combinatory Sanjay Sharma 326 views
4 votes
1 answer
40
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, 2020 in Combinatory dd 432 views
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