Show that for every n N, we have 2 > n. b) Deduce that if S...

 

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Show that for every n N, we have 2" > n. b) Deduce that if S be a subset of IN with the following properties i) 2 S for every n N ii) If n S for some n 2 then n - 1 S, then S = N. c) Show that 1 + 2 + 2 + + 2"-12" - 1. = .. d) Use parts b) and c) to show that every natural number has a binary representation that is of the form 2 +22 + 1, 12,, ik NU {0}. e) Use a parity argument to show that the binary representation of natural numbers is unique that is if n = 2 +2 + ... + 2 = 2/ +21 + +2jc, with 0 Show that for every n N, we have 2" > n. b) Deduce that if S be a subset of IN with the following properties i) 2 S for every n N ii) If n S for some n 2 then n - 1 S, then S = N. c) Show that 1 + 2 + 2 + + 2"-12" - 1. = .. d) Use parts b) and c) to show that every natural number has a binary representation that is of the form 2 +22 + 1, 12,, ik NU {0}. e) Use a parity argument to show that the binary representation of natural numbers is unique that is if n = 2 +2 + ... + 2 = 2/ +21 + +2jc, with 0