12:55 0 O & . ME . 83%4 IM_COMP20043_DiscreteStructure_... . . . Exercises: 1. Why is f not a function from R to R if a) f (x) = 1/x? b) f (x) = vx? c) f (x) = = V(x2 + 1)? 2. Determine whether f is a function from Z to R if a) f (n) = In. b) f (n) = Vn2 + 1. c) f (n) = 1/(n2 - 4). 3. Determine whether f is a function from the set of all bit strings to the set of integers if a) f (S) is the position of a 0 bit in S. b) f (S) is the number of 1 bits in S. c) f (S) is the smallest integer i such that the ith bit of S is 1 and f (S) =0 when S is the empty string, the string with no bits. 4. Find the domain and range of these functions. Note that in each case, to find the domain, determine the set of elements assigned values by the function. a) the function that assigns to each nonnegative integer its last digit b) the function that assigns the next largest integer to a positive integer c) the function that assigns to a bit string the number of one bits in the string d) the function that assigns to a bit string the number of bits in the string 5. Find the domain and range of these functions. Note that in each case, to find the domain, determine the set of elements assigned values by the function. 63 Discrete Structure 1 a) the function that assigns to each bit string the number of ones in the string minus the number of zeros in the string b) the function that assigns to each bit string twice the number of zeros in that string c) the function that assigns the number of bits left over when a bit string is split into bytes (which are blocks of 8 bits) d) the function that assigns to each positive integer the largest perfect square not exceeding this integer. 69 of 140 O12:56 0 8 7: H .1 83%4 IM_COMP20043_DiscreteStructure_.. Exercises: 1. Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. a) the negative integers b) the even integers c) the integers less than 100 d) the real numbers between 0 and 1/2 e) the positive integers less than 1,000,000,000 f ) the integers that are multiples of 7 2. Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. a) the integers greater than 10 b) the odd negative integers c) the integers with absolute value less than 1,000,000 d) the real numbers between 0 and 2 e) the set A x Z* where A = (2, 3} f ) the integers that are multiples of 10 3. Determine whether each of these sets is countable or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. a) all bit strings not containing the bit 0 b) all positive rational numbers that cannot be written with denominators less than 4 c) the real numbers not containing 0 in their decimal representation d) the real numbers containing only a finite number of 1s in their decimal representation 67 Discrete Structure 1 4. Determine whether each of these sets is countable or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. a) integers not divisible by 3 b) integers divisible by 5 but not by 7 c) the real numbers with decimal representations consisting of all 1s d) the real numbers with decimal representations of all is or 9s 5. Show that a finite group of guests arriving at Hilbert's fully occupied Grand Hotel can be given rooms without evicting any current guest. E