Question
Which of the following statements are true or false? Give reasons for your answers in the form of a short proof or counter-example, whichever is appropriate:
i) Every infinite set is an open set.
ii) The negation of p^~qis p→q.
iii) -1 is a limit point of the interval ]-2,1].
iv) The necessary condition for a function f to be integrable is that it is continuous.
The function f: R→R defined by f(x) = |x-2|+|3-x is differentiable at x = 5.
Answer :
Word Count : 532
Let's go through each statement one by one: --- ### i) Every infinite set is an open set. False. Counterexample: The set of natural numbers, \( \mathbb{N} = \{1, 2, 3, \dots\} \), is infinite, but it is not an open set in \( \mathbb{R} \) because it does not contain any open intervals around its points. In fact, for a set to be open, it must contain an open interval around each of its points, which the set of natural numbers does not. --- ### ii) The negation of \( p \land \neg q \) is \( p \to q \). True. Proof: The expression ______ ______ ______ ________ __________ _____ _________ ____ __________ ___ _________.
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Let's go through each statement one by one: --- ### i) Every infinite set is an open set. False. Counterexample: The set of natural numbers, \( \mathbb{N} = \{1, 2, 3, \dots\} \), is infinite, but it is not an open set in \( \mathbb{R} \) because it does not contain any open intervals around its points. In fact, for a set to be open, it must contain an open interval around each of its points, which the set of natural numbers does not. --- ### ii) The negation of \( p \land \neg q \) is \( p \to q \). True. Proof: The expression ______ ______ ______ ________ __________ _____ _________ ____ __________ ___ _________.
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Related Questions
कॉशी समाकल परीक्षण द्वारा निम्नलिखित मूल्यांकन कीजिए :
Let x and y be two real numbers such that x < y.Show that there exists an irrational number λ such that
x < λ < y.
Give one example each for the following. Justify your choice of examples.
i) A sequence which is divergent.
ii) A set which is neither open nor closed.
iii) A compact set.
iv) A set which has no limit point
Apply the Cauchy’s integral test to evaluate:
Find the supremum and infimum of the set:
समुच्चय का उच्चक और निम्नक ज्ञात कीजिए।
A set which has no limit point
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