Question
(7) List all isomorphisms f : Z2 X Z2 ~ + Z2 X Z2. List all isomorphisms 9 : Z2 X Z5 3 Z2 X Z5.
(7) List all isomorphisms f : Z2 X Z2 ~ + Z2 X Z2. List all isomorphisms 9 : Z2 X Z5 3 Z2 X Z5.
Answers
Let f be an ordered field and x,y,z in F.
Prove that if x<0 and y<z, then xy>xz.
You're taking quite a bit of partials here. So first one let's go ahead and take the partial respect to X first. Could just three X squared. Fly to the fifth C. Seventh plus why squared then let's take the partial with respect to why afterwards. And so then that will give us 15 X squared Y. To the fourth. As we're driving this part here. See to the seventh plus two white. Right? That's your answer for this first part here. Okay, so for the next part we're gonna have to take the partial with respect to why? First It's going to be equal to five X. Two third, right to the fourth seat seven bus two X. Y plus three Y squared C. And then take partial after that with respect to Z. That's going to be equal to 35. Next to the third, Y to the fourth C. To the sixth. Driving. Just as either this one's going to go to zero. This one's just that's the right there leaving three west. Great. As a constance, we have plus three. Why squared? So that's the area has had a party. Yeah. Come on to see So again we have half partial with respect to X. Was three X squared 55 See to the seventh plus Y squared. Going to take that, take the partial respect, dizzy afterwards. Just really cool. So then we're just arriving this and then this one is going to go to zero here. So we have seven times three is 21 X. Squared by the fifth. C. To the sixth. That's her answer to part. See going on to a party here are partial respect as you first. That was equal to seven X. Cubed by fifth C. Sixth plus what's the third? If we take the partial with respect to Z. Again. So just arrived this year, everything else is constant. Six times seven is 42 next to the third, went to the fifth, C. To the fifth. This goes to zero. So this is her answer here. Pretty okay. Next for each. So we've got the partial with respect to Z. Right there from what we had earlier. So seven X. To the 3rd. 5th C. 60 plus Y. To the third. We're going to take partial with respect to why. Afterwards it's going to drive this one here. So five times 7 35 X. Third to the fourth C. Sixth and then deriving by two thirds of its plus three Y. Squared. Depriving time with respect to why only again? So partial to why? So you got four times 14 times 35 which is 1 40 next to the third. One to the third, dizzy to the sixth and then driving three Y. Square which is plus six white. So this is her answer for pretty okay. Next for F. Do the partial with respect. So we have partial with respect to X. Three X. Squared. Y. 50 C. Seventh plus Y squared. And we do the partial with respect to X. Again that's just arriving this part of it leave everything else constant. So that's going to be six X. Right to the fifth to the seventh. And then this goes to zero. So next component of this. So with the partial with respect to why now So then that's going to derive the Y. 55 times six is 30 X. Y. Fourth. See to the seventh start. So there moving on too preachy so that is so department respect to Z. And then why? So we're going to pull that from what we had earlier from party. And so that one was 35 next to the third right to the fourth. C. To the sixth plus three Y. Squared. And then we're going to take that and then to the partial with respect to act here. And so that is going to just derive this part here and this one's going to go to zero. So three times 1 35. 3 times 35 is one of five X. Squared Y. Fourth C. To sixth. That's her answer to parch either. Okay. Thanks. Part let's look at her age. That's gonna be yes. All right. So we have pressure with respect to X. X. Y. Is what we had earlier from from a part of pull that in here again. So we've got part F. That was 30 X. Y. To the fourth. See to the seventh. And if we take now from here partial with respect to why After that retrospect is easy now. So then this is going to drive just disease. Seventh, that's to 10 X. My fourth see to the sixth. This is our answer here.
We have given a question. So they're the square minus toe that minus 12 by visitors were minus 49 in tow. Foods every square, minus one by who's that? This square plus five. That plus two in tow to the despair minus 13. Said by in a seven by by does that is square minus seven said plus three. And we need to multiply as indicated. So here we're using Ah, rule a baby in tow. See, Buddy, it can be multi play as in to see by being today. So from use after using this rule, begin right. This one has to the square minus two Zed minus 12 in the four that is square minus one in tow. So that s where minus 13 said minus seven by there is square minus 49. Was that plus two into Tuesday This square minus seven, X plus three. So here we need to factor Rise first. So after fact aeration, we can write this Termez this time s two into. Was that place to in toes that minus three into that buenos seven in tow. So that the minus one. So that plus one it is the fact arise Centam factor terms and in the denominator that Fleiss tool Zed minus three. Who said minus one zed X squared, minus full night. No, we're just cancelling the formal TEM. So after cancelling week and I do in tow, they're two minus seven. Does that plus one by that is square minus whole night. No, we can diet this Thomas doing toe that minus seven does that less fun. And in the denominator part, we are using a formula so we can day that close sour into that minus seven. Begin cancel this toe so it will become go into to said plus one by read plus seven.
Hello. Real question. Envisages when that F B and ordered field and X. So I said enough. Okay. It has also given that if X less than zero and why less than that then we need to prove that X. Y greater than access it. So let us get to hear that if access less than zero, this can be written as minus Act should be greater than zero. Okay, now here, if y is less than that so Zach minus Y should be greater than zero. Okay, no, these two have become positive quantities. Some multiplication of two positive quantities should be always positive, should always be positive. So we stretch it as minus X. Which is a positive quantity. Now into that minus Y. Which is again a positive wants to know should be positive. Let us open the bracket minus X. Z bless X. Y should be positive. Let us add except to both the sides will be having X way this is minus exceed all. It is minus except plus X. Y. And we are adding acceptable the sides greater than exit. So these two will become zero. So from here we are getting X. Y greater than X zet. So this is the thing we need to prove. Thank you.
Hello. Real question. Envisages when that F B and ordered field and X. So I said enough. Okay. It has also given that if X less than zero and why less than that then we need to prove that X. Y greater than access it. So let us get to hear that if access less than zero, this can be written as minus Act should be greater than zero. Okay, now here, if y is less than that so Zach minus Y should be greater than zero. Okay, no, these two have become positive quantities. Some multiplication of two positive quantities should be always positive, should always be positive. So we stretch it as minus X. Which is a positive quantity. Now into that minus Y. Which is again a positive wants to know should be positive. Let us open the bracket minus X. Z bless X. Y should be positive. Let us add except to both the sides will be having X way this is minus exceed all. It is minus except plus X. Y. And we are adding acceptable the sides greater than exit. So these two will become zero. So from here we are getting X. Y greater than X zet. So this is the thing we need to prove. Thank you.