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4.18. Prove the proposition Vx € R, Vy € R, 3z € R,(x - 2)2 < (x y)?. 4.19. Prove the proposition Vx € R, Jy,z € R, y2 < x2 <...

Question

4.18. Prove the proposition Vx € R, Vy € R, 3z € R,(x - 2)2 < (x y)?. 4.19. Prove the proposition Vx € R, Jy,z € R, y2 < x2 < 22 4.20. Disprove the proposition Vx € N, Jy eN,y < x. 4.21. Disprove the proposition Ix € N; Vy € N,y < x. 4.22. Prove or disprove the proposition Vx € N, Jy € N, X <y < (x +1)2. 4.23. Prove OT disprove the proposition Vx e %, Vy € Z, 3ez, (x < y) - (x < 2 < y). 4.24. Prove Ot disprove the proposition Vx € R, Vy € R

4.18. Prove the proposition Vx € R, Vy € R, 3z € R,(x - 2)2 < (x y)?. 4.19. Prove the proposition Vx € R, Jy,z € R, y2 < x2 < 22 4.20. Disprove the proposition Vx € N, Jy eN,y < x. 4.21. Disprove the proposition Ix € N; Vy € N,y < x. 4.22. Prove or disprove the proposition Vx € N, Jy € N, X <y < (x +1)2. 4.23. Prove OT disprove the proposition Vx e %, Vy € Z, 3ez, (x < y) - (x < 2 < y). 4.24. Prove Ot disprove the proposition Vx € R, Vy € R, 3z € R, (x < y) + (x < 2 < y): 59



Answers

Prove each statement for positive integers $n$ and $r,$ with $r \leq n$ $$\left(\begin{array}{c} n \\n-r\end{array}\right)=\left(\begin{array}{l}n \\r\end{array}\right)$$

Okay so in this problem what we're gonna need to do is find the minimum distance from H. To J. Two K. So the minimum distance from H. Two J. D. K. So we need you need to figure it out. So first we mark the points the points the marines are Following. So H. is equal to 10 and two And K. is equal to 18 and 17 so mhm. Now what we need to do is find the slope so 17 -2 over negative 14 -10 is equal to y minus 2/2 minus 10. So now we can cross multiplying so that will get us eight Y minus 16 equals 40. So why in this case is equal to seven because we plus 16 plus 16 and then divide by eight and divide by eight so we get white or seven. So now what we could do is calculate the distance from H. two J. So now thanks Something about this. So now what we can do is J2H. is equal to square root of seven minus two squared plus two minus 10 squared. So that's going to equal route 89. That's where we were 89. So now that this is from JK is going to equal yeah 18 minus two square plus 17 minus 17 minus seven squirt. So that's equal to him 356. So that means if we add the minimum distance together that's gonna from H to J. Two K. Is going to equal minimum distance of about 28.3. This is because we take route 89 Husband 3 56 and then We get 28 3. Okay, what's the answer?

In this problem of basics of content. Hillary. We have to prove each statement for positive and teasers and and our with our less than or equal to and and we have given permutation in coma and is equal to and factorial We take alleges. And so already using permanent permutation definition, we have formula for permutation and we are equal to N. Factorial A bond in minus R. Factorial comparing this to. We have value of pain and and value of our is also a. So we have N factorial upon in minus and factorial simplifying the expression we have in factorial upon zero factorial. We know value of zero. Factorial Is equal to one. So we have N. P. N equal to and victoria also we have value of averages is equal to in factorial. So we can say here alleges equal to our ages hands. Oh this will be our final answer.

In this problem of basics of content theory. We have to prove each statement for positive and desert, N N R. With our less than or equal to And and we have given problem B and combined -1 equal to B. And gold mine here for mutation in Coma. In -1 were to permutation in coma. First we take Alleges mm hmm. Find the value of LH is using the formula for permutation. NPR formula for permutation and pictorial upon in minus R factorial. Putting the value of Alleges NPN -1. We have and pictorial upon in minors in minus one victoria, simplifying it. We have N factorial upon. Yeah, one factorial and we know one factorial is equal to one. So we have and Victoria's this is value of villages now we fine averages the hell and being again applying the formula for permutation in factorial Upon in -1 victoria. So we have N factorial upon Jiro factorial. We know value of zero. Factorial is equal to one. so we have value of R. S equal to N. Factorial. We can see value of Alleges is in factorial and Rhs is also in factorial. So here LHs equal to rhs so we can say hence cruel. This is our final answer yeah.


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