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Propositional and first order logic.
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propositional logic
which of the following is tautology? (¬P^(P>q))>¬q ¬(p>q)>¬q [(¬p^q)^[q>(p>q)]]>¬r Both (B) and(C) please explain in detail how to check for especially for condition (C) Because “r” is only in RHS but not in LHS of this implication.
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TIFR2011A12
The action for this problem takes place in an island of Knights and Knaves, where Knights always make true statements and Knaves always make false statements and everybody is either a Knight or a Knave. Two friends A and B lives in a house. The census taker (an outsider) ... B is a Knave. A is a Knave and B is a Knight. Both are Knaves. Both are Knights. No conclusion can be drawn.
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GATE201935
Consider the first order predicate formula $\phi$: $\forall x [ ( \forall z \: z \mid x \Rightarrow (( z=x) \vee (z=1))) \rightarrow \exists w ( w > x) \wedge (\forall z \: z \mid w \Rightarrow ((w=z) \vee (z=1)))]$ Here $a \mid b$ denotes ... S2: Set of all positive integers S3: Set of all integers Which of the above sets satisfy $\phi$? S1 and S2 S1 and S3 S2 and S3 S1, S2 and S3
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Linear algebra
The number of different n×n symmetric matrices with each element being 0 or 1 is
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Discrete Maths and Its Applications, 7th Edition, Kenneth Rosen
1. Which of these sentences are propositions? What are the truth values of those that are propositions? a) Boston is the capital of Massachusetts. b) Miami is the capital of Florida. c) 2+3 = 5. d) 5+7 = 10. e) x +2 ... is answer for f) as I am thinking I can answer this question or I will not answer this question is it a proposition?
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For All & There Exist property
is For All V is distributive on AND operator? 1 is ‘’there exist’’ is distributive over OR operator? 2 I have example in which this is true , but is true in general? so logically checking without using any property above four statements are correct. But is 1 and 2 always correct?
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Permutation and Combination
The total number of ways in which 5 balls of different color can be distributed among 3 persons so that each person gets at least one ball is:
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Kenneth Rosen Ex.1.3 Q.37
Express each of these statements using predicates and quantifiers. a) A passenger on an airline qualifies as an elite flyer if the passenger flies more than 25,000 miles in a year or takes more than 25 flights during that year.
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#Basic #DiscreteMathematics #KennethRosen
Suppose that there are 9 faculty members in the mathematics department and 11 in the computer science department. How many ways are there to select a committee to develop a discrete mathematics course at a school if the committee is to consist of ... use Sum rule. How to decide on Product rule or Sum rule? Please use some basic example for both the cases.
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Rosen ch1 Ex.1.6 Q10.b
What are relevant conclusion and explain rule of inference used ? "If i work ,it is either Sunny or partly sunny." " I worked last Monday or i worked last friday." " It was not Sunny on Tuesday."" It was not partly sunny on Friday "
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mathematical logic doubt
Let L(x, y) be the statement “x loves y,” where the domain for both x and y consists of all people in the world. Use quantifiers to express each of these statements. a)There is exactly one person whom everybody loves. b) There is somebody whom no one loves. c)There are exactly two people whom Lynn loves. d)There is someone who loves no one besides himself or herself.
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GATE 2019
SOLVE: 2^32 mod 5
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Feb 11
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gate 2019
Y is distributed uniformly in [1,6] 3x^2+6xY+3Y+6=0 P(Y) for 2 real roots ?
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Feb 11
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PREDICATE LOGIC DOUBT
Only Area 51 has ExtraTerresstrials A(x) = x is Area 51 E(x) = x has ExtraTerresstrials Which of the following is correct? (∀x)(A(x) > E(x)) (∀x)(E(x) > A(x)) (∀x)(A(x) <> E(x))
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GATEBOOK2019DM110
Assuming a nonempty universe, the formula $(\forall x P(x) \vee \exists y P(y))$ is equivalent to $\exists x (P(x))$ $ (\forall x P(x))$ $ \neg (\forall x P(x))$ $ \neg (\exists x (P(x))$
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Kenneth Rosen Edition7 Ch1 Ex1 QueNo16
Determine whether these biconditionals are true or false. 2 + 2 = 4 if and only if 1 + 1 = 2. 1 + 1 = 2 if and only if 2 + 3 = 4. 1 + 1 = 3 if and only if monkeys can fly. 0 > 1 if and only if 2 > 1
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Feb 10
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GATE199202,xvi
Choose the correct alternatives (more than one may be correct) and write the corresponding letters only: Which of the following is/are a tautology? $a \vee b \to b \wedge c$ $a \wedge b \to b \vee c$ $a \vee b \to \left(b \to c \right)$ $a \to b \to \left(b \to c \right)$
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TIFR2010A4
If the bank receipt is forged, then Mr. M is liable. If Mr. M is liable, he will go bankrupt. If the bank will loan him money, he will not go bankrupt. The bank will loan him money. Which of the following can be concluded from the above statements? Mr. M is liable The receipt is not forged Mr. M will go bankrupt The bank will go bankrupt None of the above
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Feb 9
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GATEBOOK2019DM112
Translate the following logic statement to English where, $A(x)$: $x$ is African, $F(x,y)$: x and y are friends. The universe for $x$ and $y$ is all the people in the world. $\forall x \exists y((A(x) \vee (F(x,y)))$ Every African has some African friend Every person who is not African has at least one friend Every person who has friend is not African None of these
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Feb 8
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20
GATEBOOK2019DM113
Consider the following statement $ \exists x \: \exists y (\text{PARENT}(x, \text{Ramu}) \wedge \text{PARENT}(y, \text{Ramu}))$ where $\text{PARENT}(x,y)$ means $x$ is a parent of $y.$ Which of the following statement is true about ... order logic statement ? Ramu has at least one parent Ramu has at least two parents Ramu has at most one parent Ramu has at most two parents
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21
#DiscreteMathematics #Rosen #BinomialTheorem
$\sum_{k=0}^{\propto }\left ( \sum_{j=0}^{k}1 \right )x^{k} = \sum_{k=0}^{\propto}(k+1)x^{k}$ How LHS=RHS here?
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7
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22
GATE2017211
Let $p, q, r$ ... $(\neg p \wedge r) \vee (r \rightarrow (p \wedge q))$
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Feb 8
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#DiscreteMathematics #Rosen
“For every person x, if person x is a student in this class then x has studied Calculus.” S(x):Person x in the class C(x):x has studied Calculus. 1.$\vartheta _{x} (S(x) > C(x))$ 2.$\vartheta _{x} (S(x) \Lambda C(x))$ Which one should hold? Why and why not?
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24
#GeneralMathematics
How to calculate modulus of an exponential number?
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gatebook predicate logic practice problems
Hi Can anyone please explain this statements S1: ∀x ∃y ∀z [ x+ y = z] S2: ∃x ∀y ∃z [x + y = z] Where x, y, z are real numbers. Which of the following statement is true?
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GATE2005IT36
Let $P(x)$ and $Q(x)$ ...
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Gate CS 2019
S1=Matrix is Invertible S2=Determinant of matrix is Nonzero What is the Answer..
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gate 2019
What is the answer for the predicate logic question,in which a formula was given and asked to find the sets satisying it?
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Gate cse 2109
What is the answer for question based on predicate logic ?
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Fundamentals of Discrete Mathematics
Write the symbolic form? “If the utility cost goes up or the request for the additional funding is denied then a new computer will be purchased if and only if we can show that the current computing facilities are indeed not adequate.”
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31
gate 2019
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Probability: Poisson distribution calculation vs normal probability calculation
In this question if we do simply probability calculation then it is 8/20 40% but when I am appling poisson distribution then it is 40.4%. why we are getting two different answers??
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Descrete math zeal booklet
Prove that every graph with n vertices and k components has atleast nk edges.
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How to solve this generating function Question
Given two each of p kinds of objects and one each of additional q kind of objects, in how many ways r objects can be selected? Please give a detailed solution.
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graph theory
can we say a null graph is eulerian circuit and hamiltonian circuit?
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GATE20021.8
"If $X$ then $Y$ unless $Z$" is represented by which of the following formulas in prepositional logic? ("$\neg$" is negation, "$\land$" is conjunction, and "$\rightarrow$" is implication) $(X\land \neg Z) \rightarrow Y$ $(X \land Y) \rightarrow \neg Z$ $X \rightarrow(Y\land \neg Z)$ $(X \rightarrow Y)\land \neg Z$
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GATE200831
$P$ and $Q$ are two propositions. Which of the following logical expressions are equivalent? $P ∨ \neg Q$ $\neg(\neg P ∧ Q)$ $(P ∧ Q) ∨ (P ∧ \neg Q) ∨ (\neg P ∧ \neg Q)$ $(P ∧ Q) ∨ (P ∧ \neg Q) ∨ (\neg P ∧ Q)$ Only I and II Only I, II and III Only I, II and IV All of I, II, III and IV
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GATE200627
Consider the following propositional statements: $P_1: ((A ∧ B) → C)) ≡ ((A → C) ∧ (B → C))$ $P_2: ((A ∨ B) → C)) ≡ ((A → C) ∨ (B → C))$ Which one of the following is true? $P_1$ is a tautology, but not $P_2$ $P_2$ is a tautology, but not $P_1$ $P_1$ and $P_2$ are both tautologies Both $P_1$ and $P_2$ are not tautologies
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GATE2004IT31
Let $p, q, r$ and $s$ be four primitive statements. Consider the following arguments: $P: [(¬p\vee q) ∧ (r → s) ∧ (p \vee r)] → (¬s → q)$ $Q: [(¬p ∧q) ∧ [q → (p → r)]] → ¬r$ $R: [[(q ∧ r) → p] ∧ (¬q \vee p)] → r$ $S: [p ∧ (p → r) ∧ (q \vee ¬ r)] → q$ Which of the above arguments are valid? $P$ and $Q$ only $P$ and $R$ only $P$ and $S$ only $P, Q, R$ and $S$
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madeeasy
Q.The number of ways, we can arrange 5 books in 3 shelves ________.
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