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Recent questions and answers in Set Theory & Algebra
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Hasse Doubt
what is the least upper bound of {a, b, c}?
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CMI2011B02a
Let $G$ be a connected graph. For a vertex $x$ of $G$ we denote by $G−x$ the graph formed by removing $x$ and all edges incident on $x$ from $G$. $G$ is said to be good if there are at least two distinct vertices $x, y$ in $G$ such that both $G − x$ and $G − y$ are connected. Show that for any subgraph $H$ of $G$, $H$ is good if and only if $G$ is good.
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Made Easy Test Series:Lattice
The number of totally ordered set compatible to the given POSET are __________
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ISI2018MMA15
Let $G$ be a finite group of even order. Then which of the following statements is correct? The number of elements of order $2$ in $G$ is even The number of elements of order $2$ in $G$ is odd $G$ has no subgroup of order $2$ None of the above.
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Discrete mathematics #TEST_BOOK
I Have doubt about the language. Is it asking about the sum of elements if we make the GBL set for the given lattice .
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Made Easy Test Series:Discrete MathematicsPoset
Consider the following Posets: $I)\left ( \left \{ 1,2,5,7,10,14,35,70 \right \},\leq \right )$ $II)\left ( \left \{ 1,2,3,6,14,21,42 \right \},/ \right )$ $III)\left ( \left \{ 1,2,3,6,11,22,33,66 \right \},/ \right )$ Which of the above poset are isomorphic to $\left ( P\left ( S \right ),\subseteq \right )$ where $S=\left \{ a,b,c \right \}?$
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Discrete Mathematics by Kenneth Rosen,section2.4,recursive functions
$C_{a}^{k}:\mathbb{N}^{k}\rightarrow \mathbb{N}$ I am studying discrete math from beginnings and came across this term in primitive recursive function.I don't know what $C_{a}^{k}$ means and does $\mathbb{N}$ means set of natural numbers?Someone please help me out.
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ISI2018PCBB3
An $n$variable Boolean function $f:\{0,1\}^n \rightarrow \{0,1\} $ is called symmetric if its value depends only on the number of $1’s$ in the input. Let $\sigma_n $ denote the number of such functions. Calculate the value of $\sigma_4$. Derive an expression for $\sigma_n$ in terms of $n$.
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ISI2018PCBA4
Let $A$ and $B$ are two nonempty finite subsets of $\mathbb{Z}$, the set of all integers. Define $A+B=\{a+b:a\in A,b\in B\}$.Prove that $A+B\geq A +B 1 $, where $S$ denotes the cardinality of finite set $S$.
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GATE 2019
Let U = {1, 2, ..., n} and A = {(x, X), x ∈ X and X ⊆ U}. Consider the following two statements for A. (i) A = n*$\small 2^{n1}$ (ii) A= Sigma(k=1 to n) k.(nCk) Which of the following is correct? (a) (i) only (b) (ii) only (c) Both (i) and (ii) (d) None of the above
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May 11
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11
Rosen 7e Exercise9.6 Question no27 page no631
What is the covering relation of the partial ordering {(A, B)  A ⊆ B} on the power set of S, where S = {a, b, c}? i'm getting R={(Ф, {a}), (Ф, {b}), (Ф, {c}), (Ф, {a, b}), (Ф, {b, c}), (Ф, {a, c}), (Ф, {a, b, c}), ({a}, {a, b}), ({a}, {a, c}), ({b}, ... b, c}), ({c}, {a, c}), ({c}, {b, c}), ({a, b}, {a, b, c}), ({a, c}, {a, b, c})({b, c}, {a, b, c})
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Which Statement is correct for the given sets statements
If A, B, C are three sets then which of the following is TRUE ? If ( A ∩ C ) = ( B ∩ C ) then A = B If ( A ∪ C ) = ( B ∪ C ) then A = B If ( A 𝜟 C ) = ( B 𝜟 C ) then A = B If ( A – C ) = ( B – C ) then A = B
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Kenneth Rosen Edition 7th Exercise 2.3 Question 72 (Page No. 155)
Suppose that $f$ is a function from $A$ to $B$, where $A$ and $B$ are finite sets with $A=B$. Show that $f$ is onetoone if and only if it is onto.
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May 8
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ISI2019MMA19
Let $G =\{a_1,a_2, \dots ,a_{12}\}$ be an Abelian group of order $12$ . Then the order of the element $ ( \prod_{i=1}^{12} a_i)$ is $1$ $2$ $6$ $12$
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May 8
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Shikha Mallick
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15
GATE199317
Out of a group of $21$ persons, $9$ eat vegetables, $10$ eat fish and $7$ eat eggs. $5$ persons eat all three. How many persons eat at least two out of the three dishes?
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May 5
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manohar
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16
IIT Madras MS written test 2019
Which of the following infinite sets have the same cardinality? $\mathbb{N}$ : Set of Natural numbers $\mathbb{E}$ : Set of Even numbers $\mathbb{Q}$ : Set of Rational numbers $\mathbb{R}$ : Set of Real numbers $\mathbb{N}$ and $\mathbb{E}$ $\mathbb{Q}$ and $\mathbb{R}$ $\mathbb{R}$ and $\mathbb{N}$ None of the above
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May 2
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royal shubham
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17
Allen Career Institute: Discrete Math
Let $f : A \rightarrow B$ be a bijection and let $E,F$ be subjects of $A$, Now, we consider the following statements about the function $f :$ $P : f(E \cup F) = f (E) \cup f(F)$ ... None of $P$ and $Q$ is correct I thought $Q$ is true, but answer is both true. Is both true because of bijective function or ans given incorrect?
answered
May 1
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Hirak
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18
SelfDoubt:Mathematical logic
“Every asymmetric relation is antisymmetric” Is this statement is True or False? I think it is false, because asymmetric relation never allows loops and antisymmetric relation allows loops. Am I not correct?
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Apr 27
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19
POSET self doubt
What is dual of a POSET?
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20
Rosen 7e Exercise9.5 Question no9 page no615
Suppose that $A$ is a nonempty set, and $f$ is a function that has $A$ as its domain. Let $R$ be the relation on $A$ consisting of all ordered pairs $(x, y)$ such that $f (x)=f (y)$ $a)$ Show that $R$ is an equivalence relation on $A$ $b)$ What are the equivalence classes of $R?$
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Apr 23
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srestha
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21
Bounded lattice
Can a countable infinite lattice be bounded?
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Apr 20
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Set Theory & Algebra
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Manoj Kumar Pandey
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lattice
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22
GATE 2019: Equivalent Relation
Which of the following equivalent relation of a group G? R 1 : ∀ a , b ∈ G , a R 1 b if only ∃ g ∈ G : a = g − 1 bg R 2 : ∀ a , b ∈ G , a R 2 b if only a = b –1 (a) Both R 1 and R 2 (c) R 1 (b) R 2 (d) None of these
answered
Apr 19
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relations
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23
Set Theory
A relation R on a set of positive integers is defined by (a,b) belongs to R iff a and b are relatively prime. Which of the following is true about R? a. Symmetric and Reflexive b. Symmetric and irreflexive c.Symmetric and transitive d. Symmetric and not transitive The Ans is given as (d) but I think (b) is true. Any thoughts?
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Apr 19
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24
Self doubt group theory
Is (Z+,>=) a well oerderd set ,plz explain.
answered
Apr 17
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PRANAB NANDY
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25
GATE2016228
Consider a set $U$ of $23$ different compounds in a chemistry lab. There is a subset $S$ of $U$ of $9$ compounds, each of which reacts with exactly $3$ compounds of $U$. Consider the following statements: Each compound in U \ S reacts with an odd number ... in U \ S reacts with an even number of compounds. Which one of the above statements is ALWAYS TRUE? Only I Only II Only III None.
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Apr 15
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Kenneth Rosen Edition 7th Exercise 2.3 Question 74 (Page No. 155)
Prove or disprove each of these statements about the floor and ceiling functions. $\left \lfloor \left \lceil x \right \rceil \right \rfloor = \left \lceil x \right \rceil$ for all real numbers $x.$ ... $x$ and $y.$
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Kenneth Rosen Edition 7th Exercise 2.3 Question 73 (Page No. 155)
Prove or disprove each of these statements about the floor and ceiling functions. $\left \lceil \left \lfloor x \right \rfloor \right \rceil = \left \lfloor x \right \rfloor$ for all real number $x.$ ... $x.$
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Kenneth Rosen Edition 7th Exercise 2.3 Question 69 (Page No. 155)
Find the inverse function of $f(x) = x^3 +1.$
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SomeEarth
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Kenneth Rosen Edition 7th Exercise 2.3 Question 71 (Page No. 155)
Let $S$ be a subset of a universal set $U$. The characteristic function $f_{s}$ of $S$ is the function from $U$ to the set $\left \{ 0,1 \right \}$ such that $f_{S}(x)=1$ if $x$ belongs to $S$ and $f_S(x)=0$ if $x$ does not belong to $S$. Let $A$ ... $f_{A \oplus B}(x) = f_{A}(x) + f_{B}(x) 2 f_{A}(x) f_{B}(x) $
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Kenneth Rosen Edition 7th Exercise 2.3 Question 70 (Page No. 155)
Suppose that $f$ is an invertible function from $Y$ to $Z$ and $g$ is an invertible function from $X$ to $Y$. Show that the inverse of the composition $fog$ is given by $(fog)^{1} = g^{1} o f^{1}.$
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Kenneth Rosen Edition 7th Exercise 2.3 Question 68 (Page No. 155)
Draw graphs of each of these functions. $f(x) =$ $\left \lceil 3x2 \right \rceil$ $f(x) =$ $\left \lceil 0.2x \right \rceil$ $f(x) =$ $\left \lfloor 1/x \right \rfloor$ $f(x) =$ $\left \lfloor x^2 \right \rfloor$ ... $f(x) =$ $\left \lfloor 2\left \lceil x/2 \right \rceil +1/2\right \rfloor$
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Kenneth Rosen Edition 7th Exercise 2.3 Question 67 (Page No. 155)
Draw graphs of each of these functions. $f(x) =$ $\left \lfloor x+1/2 \right \rfloor$ $f(x) =$ $\left \lfloor 2x+1 \right \rfloor$ $f(x) =$ $\left \lceil x/3 \right \rceil$ $f(x) =$ $\left \lceil 1/x \right \rceil$ ... $f(x) =$ $\left \lceil \left \lfloor x12 \right \rfloor + 1/2\right \rceil$
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Kenneth Rosen Edition 7th Exercise 2.3 Question 66 (Page No. 155)
Draw the graph of the function $f(n) =$ $\left \lceil x \right \rceil +\left \lceil x/2 \right \rceil$ from $R$ to $R$
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Kenneth Rosen Edition 7th Exercise 2.3 Question 65 (Page No. 155)
Draw the graph of the function $f(n) =$\left \lfloor x \right \rfloor +\left \lfloor x/2 \right \rfloor$ from $R$ to $R$
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Kenneth Rosen Edition 7th Exercise 2.3 Question 64 (Page No. 155)
Draw the graph of the function $f(n) =$\left \lfloor x/2 \right \rfloor$ from $R$ to $R$
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Kenneth Rosen Edition 7th Exercise 2.3 Question 63 (Page No. 155)
Draw the graph of the function $f(n) =$\left \lfloor 2x \right \rfloor$ from $R$ to $R$
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Kenneth Rosen Edition 7th Exercise 2.3 Question 62 (Page No. 155)
Draw the graph of the function $f(n) = 1n^2$ from $Z$ to $Z$
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Kenneth Rosen Edition 7th Exercise 2.3 Question 61 (Page No. 155)
Data are transmitted over a particular Ethernet network in blocks of $1500$ octets (blocks of $8$ bits). How many blocks are required to transmit the following amounts of data over this Ethernet network? (Note that a byte is a synonym ... $1.544$ $\text{megabytes}$ of data $45.3$ $\text{megabytes of}$ data
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Kenneth Rosen Edition 7th Exercise 2.3 Question 60 (Page No. 155)
How many ATM cells (described in Example 28) can be transmitted in $10$ seconds over a link operating at the following rates? $128$ kilobits per second ($1$ kilobit= $1000$ bits) $300$ kilobits per second $1$ megabit per second ($1$ megabit=$1,000,000$ bits)
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Kenneth Rosen Edition 7th Exercise 2.3 Question 59 (Page No. 155)
How many bytes are required to encode $n$ bits of data where $n$ equals $7$ $17$ $1001$ $28800$
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