दिखाइए कि समुच्चय :
विवृत नहीं है।
हमारे पास समुच्चय है: $$ S = \left\{\frac{1}{n} + \frac{1 + (-1)^n}{3} : n \in \mathbb{N} \right\}. $$ ध्यान दें कि $(-1)^n$ पर निर्भर करता है कि $n$ सम है या विषम। 1. यदि $n$ विषम है ($n = 1, ____ ______ ______ _____ _______ _______ __________ ________ _________ ___ ____ ______.
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Disprove the statement:
by providing a suitable counter-example.
रीमान समाकलन से परिकलित कीजिए।
निम्नलिखित कथन तथा उनके निषेध को तर्कसंगत प्रमात्रकों का प्रयोग करते हुए आप किस प्रकार प्रस्तुत करेंगे? निषेध को शब्दों में भी दीजिए।
निम्नलिखित श्रेणियों के अभिसरण की जाँच कीजिए :
(i)
(ii)
A set which has no limit point
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.
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.