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TIFR2015B1
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Let $X$ be a proper closed subset of $[0, 1]$. Which of the following statements is always true?
The set $X$ is countable.
There exists $x \in X$ such that $X ∖ \left \{ x \right \}$ is closed
The set $X$ contains an open interval.
None of the above.
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Dec 20, 2015
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Set Theory & Algebra
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makhdoom ghaya
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What option B says here ??
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TIFR2015MathsB14
Let $G$ be a group. Suppose $G= p^{2}q$, where $p$ and $q$ are distinct prime numbers satisfying $q ≢ 1 \mod p$. Which of the following is always true? $G$ has more than one $p$Sylow subgroup. $G$ has a normal $p$Sylow subgroup. The number of $q$Sylow subgroups of $G$ is divisible by $p$. $G$ has a unique $q$Sylow subgroup.
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TIFR2015MathsB12
Let $X \subset \mathbb{R}$ and let $f,g : X \rightarrow X$ be a continuous functions such that $f(X)\cap g(X)=\emptyset$ and $f(X)\cup g(X)= X$. Which one of the following sets cannot be equal to $X$? $[0, 1]$ $(0, 1)$ $[0, 1)$ $\mathbb{R}$
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Dec 21, 2015
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Set Theory & Algebra
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TIFR2015MathsB9
Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a continuous function and $A \subset \mathbb{R}$ be defined by $A=\left\{ y \in \mathbb{R}:y = \lim_{n \rightarrow \infty} f(x_{n}), \text {for some sequence} x_{n} \rightarrow +\infty\right\}$ Then the set A is necessarily. $A$ connected set $A$ compact set $A$ singleton set None of the above
asked
Dec 21, 2015
in
Set Theory & Algebra
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makhdoom ghaya
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111
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tifrmaths2015
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TIFR2015MathsB8
For a group $G$, let $F(G)$ denote the collection of all subgroups of $G$. Which one of the following situations can occur ? $G$ is finite but $F(G)$ is infinite. $G$ is infinite but $F(G)$ is finite. $G$ is countable but $F(G)$ is uncountable. $G$ is uncountable but $F(G)$ is countable.
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Dec 21, 2015
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Set Theory & Algebra
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makhdoom ghaya
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tifrmaths2015
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