Question 4. a) Consider a subset X an order relation S'on N by m n m...

 

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Question 4. a) Consider a subset X an order relation S'on N by m≤ n m | n for m, n N. (Recall that m | n when n = mk for some k € N). = Question 5. Consider the set (2, 4, 5, 7, 8, 9, 10, 21) of the set of natural numbers N. Define i) Find all the maximal and minimal elements of (X, <). ii) Find the supremum and infimum of X. b) Let X and Y be sets. Prove that X is equipotent to a subset of Y if and only if there exists an injective function f : XY. [4,3,8] = {ff is an injective function from a subset of X to a subset of Y}. Define an ordered relation on f as follows: for f.gen, f≤g if f≤g, where f g if for every (x, y) = f, (x,y) € g. Clearly ≤ is an ordered relation on . a) Show that is not empty. b) Let I be a non-empty chain of 9. Let h := U₁. fer i) Show that h is a function from a subset of X to a subset of Y. ii) Show that h E, and it is an upper bound of I. c) Conclude that has a maximal element. [2,5,8,5] Question 4. a) Consider a subset X an order relation S'on N by m≤ n m | n for m, n N. (Recall that m | n when n = mk for some k € N). = Question 5. Consider the set (2, 4, 5, 7, 8, 9, 10, 21) of the set of natural numbers N. Define i) Find all the maximal and minimal elements of (X, <). ii) Find the supremum and infimum of X. b) Let X and Y be sets. Prove that X is equipotent to a subset of Y if and only if there exists an injective function f : XY. [4,3,8] = {ff is an injective function from a subset of X to a subset of Y}. Define an ordered relation on f as follows: for f.gen, f≤g if f≤g, where f g if for every (x, y) = f, (x,y) € g. Clearly ≤ is an ordered relation on . a) Show that is not empty. b) Let I be a non-empty chain of 9. Let h := U₁. fer i) Show that h is a function from a subset of X to a subset of Y. ii) Show that h E, and it is an upper bound of I. c) Conclude that has a maximal element. [2,5,8,5]

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Answer 1 a Let X 159101217 be a set of integers i The greatest element in X is 17 and the least element in X is 1 ii The supremum of X is 17 and the infimum of X is 1 b Let X and Y be sets Prove that ... View the full answer