Discrete Mathematics WEN-CHING LIEN Department of Mathematics National Cheng Kung University 2008 WEN-CHING LIEN Discrete Mathematics 7.1: Relations Revisited: Properties of Relations Definition (7.1) For sets A, B, any subset of A × B is called a (binary) relation from A to B. Any subset of A × A is called a (binary) relation on A. WEN-CHING LIEN Discrete Mathematics 7.1: Relations Revisited: Properties of Relations Definition (7.1) For sets A, B, any subset of A × B is called a (binary) relation from A to B. Any subset of A × A is called a (binary) relation on A. WEN-CHING LIEN Discrete Mathematics Example (7.1) a) Define the relation R on the set Z by aRb, or (a, b) ∈ R, if a ≤ b. This subset of Z × Z is the ordinary ”less than or equal to” relation on the set Z, and it can also be defined on Q or R, but not C. b) Let n ∈ Z +. For x, y ∈ Z , the modulo n relation R is defined by xRy if x − y is a multiple of n.With n = 7, we find, for instance, that 9R2, −3R11, (14, 0) ∈ R, but 3|R7 (that is , 3 is not related to 7). ...continued WEN-CHING LIEN Discrete Mathematics Example (7.1) a) Define the relation R on the set Z by aRb, or (a, b) ∈ R, if a ≤ b. This subset of Z × Z is the ordinary ”less than or equal to” relation on the set Z, and it can also be defined on Q or R, but not C. b) Let n ∈ Z +. For x, y ∈ Z , the modulo n relation R is defined by xRy if x − y is a multiple of n.With n = 7, we find, for instance, that 9R2, −3R11, (14, 0) ∈ R, but 3|R7 (that is , 3 is not related to 7). ...continued WEN-CHING LIEN Discrete Mathematics Example (7.1) a) Define the relation R on the set Z by aRb, or (a, b) ∈ R, if a ≤ b. This subset of Z × Z is the ordinary ”less than or equal to” relation on the set Z, and it can also be defined on Q or R, but not C. b) Let n ∈ Z +. For x, y ∈ Z , the modulo n relation R is defined by xRy if x − y is a multiple of n.With n = 7, we find, for instance, that 9R2, −3R11, (14, 0) ∈ R, but 3|R7 (that is , 3 is not related to 7). ...continued WEN-CHING LIEN Discrete Mathematics Example (7.1) a) Define the relation R on the set Z by aRb, or (a, b) ∈ R, if a ≤ b. This subset of Z × Z is the ordinary ”less than or equal to” relation on the set Z, and it can also be defined on Q or R, but not C. b) Let n ∈ Z +. For x, y ∈ Z , the modulo n relation R is defined by xRy if x − y is a multiple of n.With n = 7, we find, for instance, that 9R2, −3R11, (14, 0) ∈ R, but 3|R7 (that is , 3 is not related to 7). ...continued WEN-CHING LIEN Discrete Mathematics Example (7.1 continued) c) For the universe U = {1, 2, 3, 4, 5, 6, 7} consider the (fixed) set C ⊆ U where C = {1, 2, 3, 6}.Define the relation R on P(U) by ARB when A ∩ C = B ∩ C. Then the sets {1, 2, 4, 5} and {1, 2, 5, 7} are related since {1, 2, 4, 5}∩ C = {1, 2} = {1, 2, 5, 7}∩ C. Likewise we find that X = {4, 5} and Y = {7} are so related because X ∩ C = ∅ = Y ∩ C. However, the sets S = {1, 2, 3, 4, 5} and T = {1, 2, 3, 6, 7} are not related. Since S ∩ C = {1, 2, 3} 6= {1, 2, 3, 6} = T ∩ C. WEN-CHING LIEN Discrete Mathematics Example (7.1 continued) c) For the universe U = {1, 2, 3, 4, 5, 6, 7} consider the (fixed) set C ⊆ U where C = {1, 2, 3, 6}.Define the relation R on P(U) by ARB when A ∩ C = B ∩ C. Then the sets {1, 2, 4, 5} and {1, 2, 5, 7} are related since {1, 2, 4, 5}∩ C = {1, 2} = {1, 2, 5, 7}∩ C. Likewise we find that X = {4, 5} and Y = {7} are so related because X ∩ C = ∅ = Y ∩ C. However, the sets S = {1, 2, 3, 4, 5} and T = {1, 2, 3, 6, 7} are not related. Since S ∩ C = {1, 2, 3} 6= {1, 2, 3, 6} = T ∩ C. WEN-CHING LIEN Discrete Mathematics Example (7.1 continued) c) For the universe U = {1, 2, 3, 4, 5, 6, 7} consider the (fixed) set C ⊆ U where C = {1, 2, 3, 6}.Define the relation R on P(U) by ARB when A ∩ C = B ∩ C. Then the sets {1, 2, 4, 5} and {1, 2, 5, 7} are related since {1, 2, 4, 5}∩ C = {1, 2} = {1, 2, 5, 7}∩ C. Likewise we find that X = {4, 5} and Y = {7} are so related because X ∩ C = ∅ = Y ∩ C. However, the sets S = {1, 2, 3, 4, 5} and T = {1, 2, 3, 6, 7} are not related. Since S ∩ C = {1, 2, 3} 6= {1, 2, 3, 6} = T ∩ C. WEN-CHING LIEN Discrete Mathematics Example (7.1 continued) c) For the universe U = {1, 2, 3, 4, 5, 6, 7} consider the (fixed) set C ⊆ U where C = {1, 2, 3, 6}.Define the relation R on P(U) by ARB when A ∩ C = B ∩ C. Then the sets {1, 2, 4, 5} and {1, 2, 5, 7} are related since {1, 2, 4, 5}∩ C = {1, 2} = {1, 2, 5, 7}∩ C. Likewise we find that X = {4, 5} and Y = {7} are so related because X ∩ C = ∅ = Y ∩ C. However, the sets S = {1, 2, 3, 4, 5} and T = {1, 2, 3, 6, 7} are not related. Since S ∩ C = {1, 2, 3} 6= {1, 2, 3, 6} = T ∩ C. WEN-CHING LIEN Discrete Mathematics Example (7.1 continued) c) For the universe U = {1, 2, 3, 4, 5, 6, 7} consider the (fixed) set C ⊆ U where C = {1, 2, 3, 6}.Define the relation R on P(U) by ARB when A ∩ C = B ∩ C. Then the sets {1, 2, 4, 5} and {1, 2, 5, 7} are related since {1, 2, 4, 5}∩ C = {1, 2} = {1, 2, 5, 7}∩ C. Likewise we find that X = {4, 5} and Y = {7} are so related because X ∩ C = ∅ = Y ∩ C. However, the sets S = {1, 2, 3, 4, 5} and T = {1, 2, 3, 6, 7} are not related. Since S ∩ C = {1, 2, 3} 6= {1, 2, 3, 6} = T ∩ C. WEN-CHING LIEN Discrete Mathematics Definition (7.2) A relation R on a set A is called reflexive if for all x ∈ A, (x, x) ∈ R. Example (7.4) For A = {1, 2, 3, 4}, a relation R ⊆ A × A will be reflexive if and only if R ⊇ {(1, 1), (2, 2), (3, 3), (4, 4)}. Consequently, R1 = {(1, 1), (2, 2), (3, 3)} is not a reflexive relation on A, whereas R2 = {(x, y)|x, y ∈ A, x ≤ y} is reflexive on A. WEN-CHING LIEN Discrete Mathematics Definition (7.2) A relation R on a set A is called reflexive if for all x ∈ A, (x, x) ∈ R. Example (7.4) For A = {1, 2, 3, 4}, a relation R ⊆ A × A will be reflexive if and only if R ⊇ {(1, 1), (2, 2), (3, 3), (4, 4)}. Consequently, R1 = {(1, 1), (2, 2), (3, 3)} is not a reflexive relation on A, whereas R2 = {(x, y)|x, y ∈ A, x ≤ y} is reflexive on A. WEN-CHING LIEN Discrete Mathematics Definition (7.2) A relation R on a set A is called reflexive if for all x ∈ A, (x, x) ∈ R. Example (7.4) For A = {1, 2, 3, 4}, a relation R ⊆ A × A will be reflexive if and only if R ⊇ {(1, 1), (2, 2), (3, 3), (4, 4)}. Consequently, R1 = {(1, 1), (2, 2), (3, 3)} is not a reflexive relation on A, whereas R2 = {(x, y)|x, y ∈ A, x ≤ y} is reflexive on A. WEN-CHING LIEN Discrete Mathematics Definition (7.3) Relation R on set A is called symmetric if (x, y) ∈ R ⇒ (y, x) ∈ R, for all x, y ∈ A. Example (7.6) With A = {1, 2, 3}, we have: a) R1 = {(1, 2), (2, 1), (1, 3), (3, 1)} a symmetric, but not reflexive, relation on A; b) R2 = {(1, 1), (2, 2), (3, 3), (2, 3)} a reflexive, but not symmetric, relation on A; c) R3 = {(1, 1), (2, 2), (3, 3)} and R4 = {(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)}, two relations on A that are both reflexive and symmetric; and d) R5 = {(1, 1), (2, 2), (3, 3)}, a relation on A that is neither reflexive nor symmetric. WEN-CHING LIEN Discrete Mathematics Definition (7.3) Relation R on set A is called symmetric if (x, y) ∈ R ⇒ (y, x) ∈ R, for all x, y ∈ A. Example (7.6) With A = {1, 2, 3}, we have: a) R1 = {(1, 2), (2, 1), (1, 3), (3, 1)} a symmetric, but not reflexive, relation on A; b) R2 = {(1, 1), (2, 2), (3, 3), (2, 3)} a reflexive, but not symmetric, relation on A; c) R3 = {(1, 1), (2, 2), (3, 3)} and R4 = {(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)}, two relations on A that are both reflexive and symmetric; and d) R5 = {(1, 1), (2, 2), (3, 3)}, a relation on A that is neither reflexive nor symmetric.