Question

Please explain your steps as you go along, I'm having trouble understanding.

Problem 4. (20 points) For any real number c, define [c] - read as ceiling - as an integer d such that 3z with 0 <2<1 and crd - 2. Assume, as a premise, that [c] exists for every real c. a) (10 points) Prove that for any c, [c] is unique. b) (10 points) Prove that Va, b: (a + b) is equal to [a] + [6] or [a] + [6] - 1

Answer 1

@ suppose 'Tejadi 4 rel=de. claim; di-da! rej zdi > a z with of all such that c-d, Zi [c] =dq =) 7 22 with os 22.41 such that codz.-22. . . . ..dy Zdz 7222) ddr = 7-22. single of thize Assume without loss of generality that. ,. ?17122 .-) Z-2270 2. &;-;77 = 49-72 a di adz.= ?i=z2 70. idadi is an integer and olz, Z ( 1 ) o į Z-7221 . & dridza Zoz, is an integer =) .71-22=0 =)Z = Z2 sidhada 5) Te] is uniques :

② let rałam, 167:n, Tatbl=t I z such that a=m.z . . J Zg suchah and ... Ozzie that ,b=n-zq. 22 le O's 2241 atba mith-z-zz. ..☺ » atba nth-(Z+22). o&G(1 402 zq 21 .70€ zitza C.2. Suppose, os Z+ZqLI.. . Ther ② gives Tatot amth=[at+167 let 12 Z+Z2 <2. Naukrtana (tty. -) 777.29-12 Zg: :. Is Z+Zq <2 =) 04Z+Z2-1(1 » 02 Zzcl. : ☺ gives atba in the 1-73 who with oeZz(! => [atbl .= mtu-la falt 167-1.