Very confused...thanks you in advance for the help!
1. (30 points) Similar to Exercise at end of Sec 2.2 56 and 58 (Pg. 137) Suppose U be the universal set. Then determine the bit strings for the following: a. An empty set b. Universal set,U c. Symmetric difference of two sets which are subset of universal set, U (The symmetric difference of A and B, is the set containing those element in either A or B, but not in both A and B.) d. Union of n sets that all are subsets of the universal set, U e. Intersection of n sets that all are subsets of the universal set, U Define any variables (such as name of the sets, size of the universal set or the bitstring length, etc), state any assumptions and provide brief justification or explanation of the derivation of each of the above bit strings. (Provide general case answer and not example specific answer for all including justification) 2. (40 points) Exercise at end of Sec 2.3 - 22 (Pg. 153) Determine whether each of these functions is a bijection from R to R. Justify your answer a. fx)3x +4 )--32 +7 3. (30 points) Exercise at end of Sec 23- 74 (Pg.155) Prove or disprove each of these statements about the floor and ceiling functions. Clearly state any assumptions, the steps involved in the proof. (You can use counterexamples to disprove but do not use examples to prove.) a. (c in Book) r r x/2 1 /21-r x14 1 for all real numbers x b. (d in Book) L vf x 1-LVX for all positive real numbers x. c' (e in Book) Lx+LY+Lx+y-L2xjtL2yfor all real numbers x and y Bonus (10 points) Exercise at end of Sec 2.3 - 34 (Pg. 154) If fandf g are one-to-one (injective), does it follow that g is one-to-one? Justify your answer 1. (30 points) Similar to Exercise at end of Sec 2.2 56 and 58 (Pg. 137) Suppose U be the universal set. Then determine the bit strings for the following: a. An empty set b. Universal set,U c. Symmetric difference of two sets which are subset of universal set, U (The symmetric difference of A and B, is the set containing those element in either A or B, but not in both A and B.) d. Union of n sets that all are subsets of the universal set, U e. Intersection of n sets that all are subsets of the universal set, U Define any variables (such as name of the sets, size of the universal set or the bitstring length, etc), state any assumptions and provide brief justification or explanation of the derivation of each of the above bit strings. (Provide general case answer and not example specific answer for all including justification) 2. (40 points) Exercise at end of Sec 2.3 - 22 (Pg. 153) Determine whether each of these functions is a bijection from R to R. Justify your answer a. fx)3x +4 )--32 +7 3. (30 points) Exercise at end of Sec 23- 74 (Pg.155) Prove or disprove each of these statements about the floor and ceiling functions. Clearly state any assumptions, the steps involved in the proof. (You can use counterexamples to disprove but do not use examples to prove.) a. (c in Book) r r x/2 1 /21-r x14 1 for all real numbers x b. (d in Book) L vf x 1-LVX for all positive real numbers x. c' (e in Book) Lx+LY+Lx+y-L2xjtL2yfor all real numbers x and y Bonus (10 points) Exercise at end of Sec 2.3 - 34 (Pg. 154) If fandf g are one-to-one (injective), does it follow that g is one-to-one? Justify your
