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Question

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Answers

Prove each, where $x \in \mathbb{R}$ and $n \in \mathbf{Z}.$

$\left\lfloor\frac{n}{2}\right\rfloor=\frac{n-1}{2}$ if $n$ is odd.

So we want to show the ceiling of and over two is n plus one Over two went on his side that NBR by definition and see culture two. Okay. Last one for some K. In the integers. So notes. No dad. Okay. Is equal to And -1/2. I just played with this cake. A base up. We don't have The ceiling for # two. We have an art. So to K plus one. You were too which is equal to um to carry over two plus one half. Okay. We can separate this left with gave plus 1/2 just equal to. Okay. Plus. Uh huh. Okay. Is an insider. So it's stealing is just K. The ceiling of 1/2 is one. Remember we said uh to note case and minus 1/2. So you can substitute that back in and minus 1/2. Plus. Why? This gives us what and minus one last two over to which is equal to. And plus one. That's it.

Right. So for this one you're assuming that n is odd. So let's and be an odd integer. And our pride of nation. We can write in us too. Okay. Last one for something. Did you Okay. Right. Note that. Okay. is equal to N -1/2. Just manipulating the equation for us. We don't have the floor of an X squared or four. Safe water. We know what and is and is odd. So two K plus one squared over four Which is equal to four K square. That's for Okay. Last one go for. Yeah. And so this would be K squared. That's okay. Plus 1/4. We can separate them. So we have K squared the flow of case where but the floor. Okay. Plus That's Love one for case where is an integer? So in this case correct? So he's gay the flow of uh 1/4 0. So we left it case square plus k. Um If we go back to the note part which we said case and -1 over to you can substitute back that back in So and managed one of the two squared plus And -1/2. Okay so I can take out and then minus 1/2 and I'll be left with And -1/2. Last one and This is equal to and -1/2 china's and minus one Plus two for 2. So in -1 plus two. Last one. Okay And ah so we just we just multiplying this sort of the top becomes And squared -1 In the bottom two times 2 is four. And this besides that we're looking for.

So they're giving problem. We want to prove that any function F. With domain are, can be written as F equals E plus oh, where E is even an O as Ott. So we want to prove that this way of writing F is unique. So we want to know that the function F and equals E. Class. Um Well consider if we have a completely even function. So if we had a five ax equals X squared, you know, this is an even function here at E. But we also know that if we had, for example, alphabets equals zero, we know this is gonna be an odd function too. Because we have negative act of negative X. Um it will cause G Fx close negative G of negative X. We see that it's just going to be the zero function. So because of that, we know that we could just say that the O function or the odd function is zero. Same thing here. If we have x cubed, we can just say this is zero and then the odd function is the actual function itself.

You hear? The question is para magnetism of or a magnetism two plus is due to is due to presidents. Often presidents often or the electron in molecular orbital so so Or two plus, iron has 15 electrons. And after writing the molecular or vital structure of this, we will find out The para magnetism of 40 plus. Iron is due to present some odd electron in the Bye, based on why are white are vital. This is our correct answer. So let's save the options. So we see obscenity is correct.


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