# Recent questions tagged discrete-mathematics

1
Let $x=\begin{bmatrix} 3& 1 & 2 \end{bmatrix}$. Which of the following statements are true? $x^Tx$ is a $3\times 3$ matrix $xx^T$ is a $3\times 3$ matrix $xx^T$ is a $1\times 1$ matrix $xx^T=x^Tx$
2
A $n\times n$ matrix $A$ is said to be $symmetric$ if $A^T=A$. Suppose $A$ is an arbitrary $2\times 2$ matrix. Then which of the following matrices are symmetric (here $0$ denotes the $2\times 2$ matrix consisting of zeros): $A^TA$ $\begin{bmatrix} 0&A^T \\ A & 0 \end{bmatrix}$ $AA^T$ $\begin{bmatrix} A & 0 \\ 0 & A^T \end{bmatrix}$
3
Consider the following functions defined from the interval $(0,1)$ to real numbers. Which of these functions attain their maximum value in the interval $(0,1)?$ $f(x)=\frac{1}{x(1-x)}$ $g(x)=-(x-0.75)^2$ $u(x)=\sin(\frac{\pi x}{2})$ $v(x)=x^2+2x$
4
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. Let $N=\{1,2,3,...\}$ be the set of natural integers and let $f:N\times N \mapsto N$ be defined by $f(m,n)=(2m-1)*2^n.$Is $f$ injective? Is $f$ surjective? Give reasons.
5
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. Suppose $A,B$ and $C$ are $m\times m$ matrices. What does the following algorithm compute? (Here $A(i,j)$ denotes the $(i.j)^{th}$ entry of matrix $A$.) for i=1 to m for j=1 to m for k=1 to m C(i,j)=A(i,k)*B(k,j)+C(i,j) end end end
6
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. In computing, a floating point operation (flop) is any one of the following operations performed by a computer ... $c_{ij}=\displaystyle\sum^5 _{k=1} a_{ik} b_{kj}$. How does this number change if both the matrices are upper triangular?
7
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. A function $f$ from the set $A$ to itself is said to have a fixed point if $f(i)=i$ for some $i$ in $A$. Suppose $A$ is the set $\{a,b,c,d\}$. Find the number of bijective functions from the set $A$ to itself having no fixed point.
1 vote
8
The number of positive integers not exceeding $100$ that are either odd or the square of an integer is _______ $63$ $59$ $55$ $50$
9
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? $4$ $6$ $7$ $9$
1 vote
10
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 $((p \rightarrow r) \wedge (q \rightarrow r))$
11
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 ______ $23$ $99$ $4$ $7$
12
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 ______ $\forall _x (f(x) \wedge \neg p(x))$ $\exists _x (f(x) \wedge \neg p(x))$ $\neg (f(x) \wedge \neg p(x))$ $\exists _x (\neg f(x) \wedge \neg p(x))$
13
What kind of clauses are available in conjunctive normal form? Disjunction of literals Disjunction of variables Conjunction of literals Conjunction of variables
14
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 $a$
15
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
16
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
17
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
18
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
19
Solve the recurrence relation for the number of rounds in the tournament described in question $14.$
20
How many rounds are in the elimination tournament described in question $14$ when there are $32$ teams?
21
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.
22
Give a big-O estimate for the function $f$ given below if $f$ is an increasing function. $f (n) = 2f (n/3) + 4 \:\text{with}\: f (1) = 1.$
1 vote
23
Find $f (n)$ when $n = 3k,$ where $f$ satisfies the recurrence relation $f (n) = 2f (n/3) + 4 \:\text{with}\: f (1) = 1.$
24
Give a big-O estimate for the function $f$ in question $10$ if $f$ is an increasing function.
25
Find $f (n)$ when $n = 2^{k},$ where $f$ satisfies the recurrence relation $f (n) = f (n/2) + 1 \:\text{with}\: f (1) = 1.$
26
Suppose that $f (n) = f (n/5) + 3n^{2}$ when $n$ is a positive integer divisible by $5, \:\text{and}\: f (1) = 4.$ Find $f (5)$ $f (125)$ $f (3125)$
Suppose that $f (n) = 2f (n/2) + 3$ when $n$ is an even positive integer, and $f (1) = 5.$ Find $f (2)$ $f (8)$ $f (64)$ $(1024)$
Suppose that $f (n) = f (n/3) + 1$ when $n$ is a positive integer divisible by $3,$ and $f (1) = 1.$ Find $f (3)$ $f (27)$ $f (729)$
How many operations are needed to multiply two $32 \times 32$ matrices using the algorithm referred to in Example $5?$