# All Questions

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### Distribution of liquid among different capacity cans

There are three cans A,B,C. The capacities of A,B and C are 6 liters, 10 liters and 16 liters respectively. Can C contains 16 liters of milk. The milk has to be divided in them using these three cans ...
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### Show the Volterra Operator is compact using only the definition of compact

The Volterra operator $V:L^{2}[0,1]\rightarrow L^{2}[0,1]$ is defined by $(Vf)(x)=\int_0^xf(t)dt$. I am wondering if it can be shown that $V$ is compact by definition - that is, either that $V$ ...
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### What is the cardinality of the set of all infinite sequences?

The set is defined as {$(n_1, n_2,...n_k ..) | n_k \in \mathbb{N}$}. What are some approaches to finding and proving the cardinality of this set?
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### 2nd Question in introductory probability

There are the same number of red and blue people at a party. And decided to meet up the next day. 1/5 of the red people and 1/3 of all the people were missing. Calculate the probability that a ...
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### prove perfect square

Show that if $ab$ and $bc$ are perfect squares then $ac$ is a perfect square using theorem of arithmetic. I am not exactly sure how to prove this. I know that a perfect square must have even powers of ...
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### Probability and proportion from standard deviation and mean

I do not know how to start this problem-help needed-kindly help me The lifetimes of a colour TV picture tube is normally distributed, with a mean of 8 yrs and a standard deviation of 2 years. ...
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### Finding the remainder of a linear congruence

Okay so say I have $314^{420} \equiv r \pmod{1001}$ and I have to find what the remainder is, $r$ in this case. I know you could compute it by $gcd(314^{420}, 1001)$ and using EEA. But the numbers are ...
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### Prove that $\lfloor -a \rfloor=-\lceil a \rceil$

Let $\lfloor a \rfloor$ be the least element of {$k \in \Bbb Z : k\ge a$} and $\lceil a \rceil$ the greatest element of {$k \in \Bbb Z : k\le a$}. How can I prove that for all $a \in \Bbb R$, ...