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MATH 209/509 - FALL 2022 PROBLEM SET # 4 Below Z denotes the set of integers and N denotes the set of non-negative integers. (1) For the following relations R decide which of them is a function: • X = {1, 2, 3, 4}, Y = {2, 3, 4}, R = {(1, 2), (2, 3), (3, 4)} ⊂ X × Y ; • X = Z, Y = N, R = {(x, y) ∈ X × Y | |x| = |y|}; • X = Z = Y , R = {(x, y) ∈ X × Y | x = |y|}. (2) Let X = {1, 2, 3, 4} and consider a function f : X → X defined in the following way: f(1) = 4, f(2) = 1, f(3) = 4, f(4) = 2. Find • f−1({1, 4}) • f(f−1({1, 4})) • f({1, 4}) • f−1(f({1, 4})) (3) For X = {1, 2, 3, 4} and Y = {1, 2, 3} write down an example of a function f : X → Y which is • surjective but not injective; • injective but not surjective; • neither injective nor surjective. (4) Consider a function f : {1, 2} × {2, 3, 4} → {2, 3, 4} which is the projection onto the second coordinate, i.e., f((x, y)) = y. • Find f({(1, 2), (2, 3)}); • Find f({(1, 2), (2, 2)}); • Find f−1({2}); • Find f−1({3, 4}). (5) Consider a function f : Z→ Z given by f(x) = x2 − 1. Find f(N) and f−1(N). (6) For an integer x write x+4Z for the equivlaence class of x for the relation that is congruence mod 4. Consider a fuction f : {1, 2, 3, 4, 5, 6} → Z/4Z defined by f(x) = x + 4Z. Is f injective? Surjective? Find • f({1, 5}) • f−1({3 + 4Z}) • f−1({1 + 4Z, 2 + 4Z}) Can you find a subset A of Y such that f(f−1(A)) 6= A? 1 2 MATH 209/509 - FALL 2022 PROBLEM SET # 4 (7) Let f : X → Y be a function. Prove that for any subsets A and B of Y one has f−1(A∪B) = f−1(A) ∪ f−1(B). MATH 509 - FALL 2022 PROBLEM SET # 5 Below N denotes the set of all natural numbers (including zero) and Q+ denotes the set of positive rationals. (1) Let X = {1, 2, 3, 4} with a partial order denoted by �. For the different � below decide which statements are true: (a) x � y if x ≥ y. Is it true that: (i) 2 � 2? (ii) 3 � 4? (iii) 4 � 1? (iv) 3 ≺ 4? (v) 2 is an immediate predecessor of 3? (b) x � y if x divides y, Is it true that (i) 1 � 2? (ii) 1 � 3? (iii) 2 � 3? (iv) 2 ≺ 3? (v) 2 ≺ 4? (vi) 2 is an immediate predecessor of 3? (2) Let X = P({0, 1, 2, 3, 4, 5})−{0, 1, 2, 3, 4, 5} with the partial order given by A � B if A ⊂ B. • Find all the immediate successors of {1, 2}. • Find all the immediate predecessors of {1, 2}. • Find all the immediate predecessors of {4} • Find all the immediate successors of {4}. • Find the largest and the smallest element of X if they exist. • Find all the minimal and all the maximal elements of X. (3) For the following relations R on X×X decide which of them are a partial order/total order. If R is a partial order, determine the least element, the greatest element (if they exist), minimal and maximal elements. • X = {1, 2, 3, 4}, R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}, • X = {1, 2, 3}, R = {(1, 1), (1, 3), (2, 2), (3, 3)}, • X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, R = {(x, y) ∈ X ×X | x divides y}, • X = {1, 3, 5, 15}, R = {(x, y) ∈ X ×X | x divides y}, • X = P({1, 2, 3, 4})− {∅, {1}, {2}, {3}, {4}}, R = {(x, y) ∈ X ×X | x ⊂ y}. 1 2 MATH 509 - FALL 2022 PROBLEM SET # 5 (4) Consider X = N with the partial order given by the ‘less than or equal’ relation ≤. For the following sets E find the set E∗ of all upper bounds, the set E∗ of all lower bounds, supE and inf E. • E = {3, 5, 6} • E = the set of even natural numbers including 0 • E = the set of all divisors of 100. (5) Let X = Q+ denote the set of positive rationals with the partial order given by the “less than or equal” relation ≤. For the following sets E find the set E∗ of all upper bounds, the set E∗ of all lower bounds, supE and inf E, if they exist. • E = {3, 5, 6} • E = {a ∈ Q+ | a ≥ 1} • E = {a ∈ Q+ | a > 1} • E = {a ∈ Q+ | 2 ≤ a < 3}="" •="" {2−n="" ∈="" q+="" |="" n="" ∈="" n}="" •="" {a="" ∈="" q+="" |="" a2="">< 2}="" (6)="" consider="" the="" set="" y="{1," 2,="" 3,="" 4}.="" we="" equip="" its="" power="" set="" x="P(Y" )="" with="" partial="" order="" �="" given="" by="" inclusion,="" i.e.,="" a="" �="" b="" if="" a="" ⊂="" b.="" •="" find="" all="" the="" strict="" predecessors="" and="" all="" the="" strict="" successors="" of="" {1,="" 2}="" as="" well="" as="" all="" the="" immediate="" predecessors="" and="" all="" the="" immediate="" successors="" of="" {1,="" 2}.="" •="" let="" e="{{1," 2},="" {1}}="" ⊂="" p(y="" ).="" find="" the="" set="" e∗="" of="" all="" upper="" bounds,="" the="" set="" e∗="" of="" all="" lower="" bounds,="" supe="" and="" inf="" e="" if="" they="" exist.="" •="" let="" e="{{1," 2},="" {1},="" {2,="" 3}}="" ⊂="" p(y="" ).="" find="" the="" set="" e∗="" of="" all="" upper="" bounds,="" the="" set="" e∗="" of="" all="" lower="" bounds,="" supe="" and="" inf="" e="" if="" they="" exist.="" •="" let="" e="" be="" the="" subset="" of="" p(y="" )="" consisting="" of="" all="" singletons,="" i.e.,="" e="{{1}," {2},="" {3},="" {4}}.="" find="" the="" set="" e∗="" of="" all="" upper="" bounds,="" the="" set="" e∗="" of="" all="" lower="" bounds,="" supe="" and="" inf="" e="" if="" they="" exist.="" (7)="" consider="" the="" set="" a="{1," 2,="" 3,="" 4}="" and="" order="" a×a="" by="" the="" lexicographical="" order,="" i.e.,="" (x,="" y)="" �="" (x′,="" y′)="" if="" either="" •="" x="">< x′ or • x = x′ and y ≤ y′. let e = {(2, 2), (2, 3), (3, 4)} ⊂ a × a. find the set e∗ of all upper bounds, the set e∗ of all lower bounds, supe and inf e if they exist. x′="" or="" •="" x="x′" and="" y="" ≤="" y′.="" let="" e="{(2," 2),="" (2,="" 3),="" (3,="" 4)}="" ⊂="" a="" ×="" a.="" find="" the="" set="" e∗="" of="" all="" upper="" bounds,="" the="" set="" e∗="" of="" all="" lower="" bounds,="" supe="" and="" inf="" e="" if="" they="">