Logic, Set Theory and Computability [M. Coppenbarger]
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1 7 Relations (Handout) Definition 7-1: A set r is a relation from X to Y if r X Y. If X = Y, then r is a relation on X. Definition 7-2: Let r be a relation from X to Y. The domain of r, denoted dom r, is defined as dom r = { x : (x,y) r for some y Y }. The range of r, denoted ran r, is defined as ran r = { y : (x,y) r for some x X }. Definition 7-3: Let r be a relation from X to Y. Let A X. The image of A through r, denoted r(a), is defined as r(a) = { y : (x,y) r for some x A }. Let B Y. The preimage of B through r, denoted r (B), is defined as r (B) = { x : (x,y) r for some y B }. Definition 7-4: Let r be a relation from X to Y. The inverse of r, denoted r 1, is defined as r 1 = { (y,x) : (x,y) r }. Definition 7-5: Given a set X, the identity relation, denoted i X, is a relation on X and is defined as i X = { (x,x) : x X }. Definition 7-6: Let r be a relation from X to Y 1 and s a relation from Y 2 to Z. The composition of r and s, denoted s r, is a relation from X to Z and is defined by s r = { (x,z) : (x,y) r and (y,z) s for some y Y 1 Y 2 }. Definition 7-7: Let r be a relation on X. The power of r is a sequence of relations r 0, r 1, r 2, r 3, where each is defined as r 0 = i X, r 1 = r, r 2 = r r 1, and r 3 = r r 2, and so on Definition 7-8: Let r be a relation from X to Y and let A X. The restriction of r to A, denoted r A, is the relation r A = r (A Y). Definition 7-9: Let A X and let s be a relation from A to Y. If there exists a relation r from X to Y such that s = r A, then r is called an extension of s. Definition 7-10 (Properties of Relations): A relation r on X is (Property I) reflexive for all x X, (x,x) r (Property I ) irreflexive for all x X, (x,x) r (Property II) symmetric (x,y) r implies (y,x) r (Property II ) asymmetric (x,y) r implies (y,x) r (Property III) antisymmetric (x,y),(y,x) r imply x = y (Property III ) antiasymmetric (x,y),(y,x) r imply x y (Property IV) transitive (x,y),(y,z) r imply (x,z) r (Property IV ) atransitive (x,y),(y,z) r imply (x,z) r (Property V) circular (x,y),(y,z) r imply (z,x) r (Property V ) acircular (x,y),(y,z) r imply (z,x) r Definition 7-11: A relation is a quasi-ordering if it is reflexive and transitive. A quisi-ordering that is symmetric is an equivalence relation. A quasi-ordering that is anti-symmetric is an order. Definition 7-12: Let r be an equivalence relation on X. For any x X, the equivalence class of x, denoted x / r, is defined as x / r = { y : y X and (x,y) r }. Definition 7-13: The collection C is mutually disjoint if A,B C implies A B = or A = B. Definition 7-14: The collection C is a partition of a set X if C X such that «C = X and C is mutually disjoint. Theorem 7-15: Let r be an equivalence relation on X. Then the set { x / r : x X } is a partition of X.
2 Definition 7-16: Let r be an equivalence relation on X. The partition induced by r on X, denoted X / r, is defined as X / r = { x / r : x X }. Theorem 7-17: Let C be a partition of X and define a relation s on X by s = { (x,y) : x,y X and there exists A C such that x,y A }. Then s is an equivalence relation. Definition 7-18: Let C be a partition of X. The relation induced by C on X, denoted X / C, is defined by X / C = { (x,y) : x,y X and there exists A C such that x,y A }. Beginning Exercises 7-A-1. List all the relations from X = { a, b } to Y = { c }. 7-A-2. List all the relations on (a) the set 1; (b) the set 2. 7-A-3. List the ordered pairs in the relation r on 2 defined by r = { (A,B) : A,B 2 and A B }. 7-A-4. Let r be a relation from 5 to 6 defined by r = { (1,2), (1,4), (2,3), (3,1), (4,2), (4,4) }. Let A = 3. Determine each of the following. (a) dom r; (b) ran r; (c) r(a); (d) r (A); (e) r 1 ; (f) i 5 ; (g) (r r)(a); (h) r r; (i) r 3 ; (j) r A. 7-A-5. Let r be a relation from X to Y. (That is, r X Y.) Prove each of the following. (a) r (Y ) = dom r X. (b) r(x) = ran r Y. (c) r 1 is a relation from Y to X and (r 1 ) 1 = r. (d) If A X, then r(a) Y. 7-A-6. Let r be a relation from X to Y. Do any of the expressions ««r,»«r, r, or»»r simplify to dom r or ran r? 7-A-7. Let r be a relation on X. (That is, r X X.) Prove that dom r 1 = ran r and ran r 1 = dom r. Medium Exercises 7-B-1. Prove that the composition of two relations is a relation. 7-B-2. Let s be a relation from X to Y. Prove each of the following. (a) i dom s s 1 s (b) i ran s s s 1 (c) i Y s = s i X = s 7-B-3. Let s be a relation from X to Y 1 and t be a relation from Y 2 to Z with A, B X. Prove (t s)(a) = t (s (A)). 7-B-4. Let s be a relation from X to Y 1 and t be a relation from Y 2 to Z. Prove (t s) 1 = s 1 t 1. 7-B-5. Let r be a relation from X to Y 1, s be a relation from Y 2 to Z 1, and t be a relation from Y 3 to Z 2. Prove if s t, then s r t r. 7-B-6. Let r be a relation from X 1 to Y 1, s be a relation from X 2 to Y 2, and t be a relation from Y 3 to Z. Prove if r s, then t r t s.
3 7-B-7. Prove that the composition of relations is associative. That is, if r is a relation from W to X 1, s is a relation from X 2 to Y 1, and t is a relation from Y 2 to Z, then t (s r) = (t s) r. 7-B-8. Let r,s be relations on X. Determine if the following are true. If not, is one a subset of the other? Prove either case. (a) dom s r = dom r (b) ran s r = ran r 7-B-9. Let r be a relation from X to Y and let A X. Prove or disprove A = r ( r(a)). If false, prove or disprove A r ( r(a)) and r ( r(a)) A. Exercises Involving Collections and Relations 7-C-1. Let s be a relation from X to Y and suppose C X is a collection of subsets of X. Prove s («C) = «{ s(a) : A C }. 7-C-2. Let s be a relation from X to Y and suppose C X a nonempty collection of subsets of X. (a) Prove that s (»C)»{ s(a) : A C }. (b) Provide a counterexample of»{ s(a) : A C } s (»C). (c) Provide a necessary and sufficient condition (or conditions) for s (»C) =»{ s(a) : A C }. 7-C-3. Let s be a relation from X to Y and suppose C Y a collection of sets. Prove each of the following. (a) s («C) = «{ s (A) : A C } (b) If C is nonempty, then s (»C) =»{ s (A) : A C }. 7-C-4. Let R be a collection of relations from X to Y. Prove each of the following. (a) «R is a relation from X to Y and («R) 1 = «{ s 1 : s R } (b) If R is nonempty, then»r is a relation from X to Y and (»R) 1 =»{ s 1 : s R }. 7-C-5. Let r be a relation from X to Y and suppose R is a collection of relations from Y to Z. Prove each of the following. (a) («R) r = «{ s r : s R } (b) If R is nonempty, then (»R) r =»{ s r : s R }. 7-C-6. Let R be a collection of relations from X to Y and suppose t be a relation from Y to Z. Prove each of the following. (a) t («R) = «{ t s : s R } (b) If R is nonempty, then t (»R) =»{ t s : s R }. Exercises Applying Properties of Relations 7-D-1. Let r be a relation on 4 defined by r = { (0,1), (0,3), (1,2), (2,0), (3,1), (3,3) }. Which of the ten properties does r satisfy? 7-D-2. How many of the relations on the set 2 in your answer for Exercise 7-A-2(b) are (I) reflexive? (II) symmetric? (III) anti-symmetric? (IV) transitive? (V) circular? (I ) irreflexive? (II ) asymmetric? (III ) anti-asymmetric? (IV ) atransitive? (V ) acircular? 7-D-3. Let W be the set of all strings of English letters of finite length. Define a relation r on W as r = { (w 1, w 2 ) : w 1 and w 2 have no letters in common }. (a) Let S be the set containing only the strings CAT and DOG. Determine the sets:
4 (i) dom r; (ii) ran r; (iii) r(s); (iv) r (S); (v) r 1 ; (vi) i W ; (vii) (r r)(s); (viii) r r; (xi) r 3 ; (x) r S. 7-D-4. Let W be the set of all strings of English letters of finite length. Define a relation s on W as s = { (w 1, w 2 ) : w 1 and w 2 do not have the same length }. (a) Let S be the set containing only the strings CAT and DOG. Determine the sets: (i) dom s; (ii) ran s; (iii) s(s); (iv) s (S); (v) s 1 ; (vi) i W ; (vii) (s s)(s); (viii) s s; (xi) s 3 ; (x) s S. 7-D-5. Let W be the set of all strings of English letters of finite length. Define a relation t on W as t = { (w 1, w 2 ) : w 1 is longer than w 2 }. (a) Let S be the set containing only the strings CAT and DOG. Determine the sets: (i) dom t; (ii) ran t; (iii) t(s); (iv) t (S); (v) t 1 ; (vi) i W ; (vii) (t t)(s); (viii) t t; (xi) t 3 ; (x) t S. 7-D-6. Let L be the set of all lines in R 2 (that is, the xy-plane where R is the set of real numbers). Define a relation r on L as r = { ( 1, 2 ) : 1 and 2 are parallel }. (a) Let S be the set containing only the x-axis and the y-axis. Determine the sets: (i) dom r; (ii) ran r; (iii) r(s); (iv) r (S); (v) r 1 ; (vi) i L ; (vii) (r r)(s); (viii) r r; (xi) r 3 ; (x) r S. 7-D-7. Let L be the set of all lines in R 2 (that is, the xy-plane where R is the set of real numbers). Define a relation s on L as s = { ( 1, 2 ) : 1 and 2 are perpendicular }. (a) Let S be the set containing only the x-axis and the y-axis. Determine the sets: (i) dom s; (ii) ran s; (iii) s (S); (iv) s (S); (v) s 1 ; (vi) i L ; (vii) (s s)(s); (viii) s s; (xi) s 3 ; (x) s S. 7-D-8. Let L be the set of all lines in R 2 (that is, the xy-plane where R is the set of real numbers). Define a relation t on L as t = { ( 1, 2 ) : 1 and 2 intersect }. (a) Let S be the set containing only the x-axis and the y-axis. Determine the sets: (i) dom t; (ii) ran t; (iii) t(s); (iv) t (S); (v) t 1 ; (vi) i L ; (vii) (t t)(s); (viii) t t; (xi) t 3 ; (x) t S. 7-D-9. Let T be the set of all non-degenerate triangles in R 2 (that is, the xy-plane). Define a relation r on T by r = { (t 1, t 2 ) : t 1 and t 2 have an angle of the same measure }. (a) Let S be the singleton set containing only the triangle with vertices at (0,0), (1,0), and (0,1). Determine the sets: (i) dom r; (ii) ran r; (iii) r(s); (iv) r (S); (v) r 1 ; (vi) i T ; (vii) (r r)(s); (viii) r r; (xi) r 3 ; (x) r S.
5 7-D-10. Let Z be the set of integers. Define a relation r on Z by r = { (m,n) : m divides n }. [For m,n Z with m 0, we say that m divides n if there exists a unique k Z such that km = n.] (a) Let S = {2, 3}. Determine the sets: (i) dom r; (ii) ran r; (iii) r(s); (iv) r (S); (v) r 1 ; (vi) i Z ; (vii) (r r)(s); (viii) r r; (xi) r 3 ; (x) r S. 7-D-11. Let Z be the set of integers. Define a relation r on Z Z by r = { ( (a,b), (c,d) ) : a < c or [ a = c and b d ] }. (a) Let S = {(1,2), (5,7)}. Determine the sets: (i) dom r; (ii) ran r; (iii) r(s); (iv) r (S); (v) r 1 ; (vi) i Z Z ; (vii) (r r)(s); (viii) r r; (xi) r 3 ; (x) r S. 7-D-12. Let X be a set. Define the relation r on X by r = { (A,B) : A,B X and A B }. (a) Let S = {, X }. Determine the sets: (i) dom r; (ii) ran r; (iii) r(s); (iv) r (S); (v) r 1 ; (vi) i X ; (vii) (r r)(s), (viii) r r, (xi) r 3, and (x) r S. 7-D-13. Let X be a set and Y X. Define the relation r Y on X by r Y = { (A,B) : A Y = B Y }. Determine which of the ten relation properties that r Y satisfies. If a relation does not satisfy a property, then provide a counterexample to show that the property is not satisfied. Proofs are not 7-D-14. Let Q be the set of rational numbers. By Q[X] is meant the set of all polynomials in X with rational coefficients. Define a relation r on A = Q[X] Q as r = { (f, g) : f, g A and g = qf for some q A }. Determine which of the ten relation properties that r satisfies. If a relation does not satisfy a property, then provide a counterexample to show that the property is not satisfied. Proofs are not Next time Let C be the set of circles in the plane. Define the relation r on C by r = { (C 1,C 2 ) : C 1 is contained within C 2 }. Let C be the set of convex regions in the plane. Define the relation r on C by r = { (C 1,C 2 ) : C 1 is a subregion of C 2 }. Exercises Involving Proofs with Properties of Relations 7-E-1. Let s be a relation on X. Find a necessary and sufficient condition so that s satisfies the given property. Then prove that the condition is true. (a) reflexive (c) symmetric (e) antisymmetric (g) transitive (i) circular (b) irreflexive (d) asymmetric (f) antiasymmetric (h) atransitive (j) acircular (Conditions should be in terms of s along with one or more of the following: i X, X, inverse, relative complement, union, intersection, composition, subset, domain, range, image, and preimage.) 7-E-2. Let r be a relation on X. Find a necessary and sufficient condition for which the following relations are reflexive. (a) r 1 r
6 (b) r r 1 7-E-3. Let r be a relation on X. Prove each of the following. (a) r 1 r is symmetric. (b) r r 1 is symmetric. 7-E-4. Let r be a relation on X. Prove that the following are equivalent: (i) r is reflexive; (ii) r 1 is reflexive; (iii) X X r is irreflexive. 7-E-5. Let r be a nonempty relation on X. Suppose that r satisfies any two of the following properties: irreflexive, symmetric, and transitive. Prove that r cannot satisfy the third property. 7-E-6. Let s be a relation on X. Prove that if s is transitive and reflexive, then s 2 = s. Is the converse true? 7-E-7. Let r be a relation on X. Prove each of the following. (a) The relation t = r r 1 is symmetric and is the smallest symmetric relation containing r. [That is, if s is a symmetric relation such that r s t, then s = t.] (b) The relation t = r r 1 is symmetric and is the largest symmetric relation contained in r. [That is, if s is a symmetric relation such that t s r, then s = t.] For the next three exercises, let P be the set of all ten relation properties. That is, P = {reflexive, irreflexive, symmetric, asymmetric, anti-symmetric, anti-asymmetric, transitive, atransitive, circular, acircular} 7-E-8. Let s be a relation on X. For each p P, prove or disprove each of the following. (a) If s satisfies property p, then s 2 satisfies property p. (b) If s 2 satisfies property p, then s satisfies property p. 7-E-9. Let s and t be relations on X with s t. For each p P, prove or disprove each of the following. (a) If s satisfies property p, then t satisfies property p. (b) If t satisfies property p, then s satisfies property p. 7-E-10. Let s and t be relations on X and let = {,,, }. For each p P and each, prove or disprove: If s and t satisfy property p, then s t satisfies property p. Exercises Involving Equivalence Relations 7-E-1. Can be an equivalence relation? 7-E-2. Construct all equivalence relations on the set 3. 7-E-3. Which of the relations in the following exercises are equivalence relations? 7-D-3 7-D-4 7-D-5 7-D-6 7-D-7 7-D-8 7-D-9 7-D-10 7-D-11 7-D-12 Of those that are equivalence relations, describe the equivalence classes. 7-E-4. Let X be a set. Prove that the relation r = { (A, B) : A,B X and A = B } is an equivalence relation on X. 7-E-5. Let r and s be relations on X. Prove that r s is an equivalence relation iff r s = s r. 7-E-6. Let r be a relation on X. Prove that r is an equivalence relation iff (i X r 1 r 2 ) r. 7-E-7. Let s be an equivalence relation on X and t an equivalence relation on Y. Define a relation r on X Y as
7 r = { ( (x 1,y 1 ), (x 2,y 2 ) ) r : (x 1,x 2 ) s and (y 1,y 2 ) t }. Prove that r is an equivalence relation on X Y. 7-E-8. Let E be a collection of equivalence relations on a set X. (a) Prove that if E is nonempty, then»e is an equivalence relation. (b) Provide a counterexample to show that «E is not always an equivalence relation. (c) Provide a necessary and sufficient condition for «E to be an equivalence relation.
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