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).

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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).

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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).

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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.

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.

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 (7.4) For a set A, a relation R on A is called transitive if, for all x, y, z ∈ A, (x, y), (y, z) ∈ R ⇒ (x, z) ∈ R.

Example (7.7) All the relations in Examples 7.1 and 7.2 are transitive, as are the relations in Examples 7.3(c)

WEN-CHING LIEN Discrete Mathematics Definition (7.4) For a set A, a relation R on A is called transitive if, for all x, y, z ∈ A, (x, y), (y, z) ∈ R ⇒ (x, z) ∈ R.

Example (7.7) All the relations in Examples 7.1 and 7.2 are transitive, as are the relations in Examples 7.3(c)

WEN-CHING LIEN Discrete Mathematics Definition (7.5) Given a relation R on a set A , R is called antisymmetric if for all a, b ∈ A, (aRb and bRa) ⇒ a = b.

Example (7.11) For a universe U, define the relation R on P(U) by (A, B) ∈ R if A ⊆ B, for (A, B) ⊆ U. So R is the subset relation of Chapter 3 and if ARB and BRA, then we have A ⊆ B and B ⊆ A, which gives us A = B. Consequently, this relation is antisymmetric, as well as reflexive and transitive, but it is not symmetric.

WEN-CHING LIEN Discrete Mathematics Definition (7.5) Given a relation R on a set A , R is called antisymmetric if for all a, b ∈ A, (aRb and bRa) ⇒ a = b.

Example (7.11) For a universe U, define the relation R on P(U) by (A, B) ∈ R if A ⊆ B, for (A, B) ⊆ U. So R is the subset relation of Chapter 3 and if ARB and BRA, then we have A ⊆ B and B ⊆ A, which gives us A = B. Consequently, this relation is antisymmetric, as well as reflexive and transitive, but it is not symmetric.

WEN-CHING LIEN Discrete Mathematics Definition (7.5) Given a relation R on a set A , R is called antisymmetric if for all a, b ∈ A, (aRb and bRa) ⇒ a = b.

Example (7.11) For a universe U, define the relation R on P(U) by (A, B) ∈ R if A ⊆ B, for (A, B) ⊆ U. So R is the subset relation of Chapter 3 and if ARB and BRA, then we have A ⊆ B and B ⊆ A, which gives us A = B. Consequently, this relation is antisymmetric, as well as reflexive and transitive, but it is not symmetric.

WEN-CHING LIEN Discrete Mathematics Definition (7.5) Given a relation R on a set A , R is called antisymmetric if for all a, b ∈ A, (aRb and bRa) ⇒ a = b.

Example (7.11) For a universe U, define the relation R on P(U) by (A, B) ∈ R if A ⊆ B, for (A, B) ⊆ U. So R is the subset relation of Chapter 3 and if ARB and BRA, then we have A ⊆ B and B ⊆ A, which gives us A = B. Consequently, this relation is antisymmetric, as well as reflexive and transitive, but it is not symmetric.

WEN-CHING LIEN Discrete Mathematics Definition (7.6) A relation R on a set A is called a partial order, or a partial ordering relation, if R is reflexive, antisymmetric, and transitive.

Definition (7.7) An equivalent relation R on a set A is a relation that is reflexive, symmetric, and transitive.

WEN-CHING LIEN Discrete Mathematics Definition (7.6) A relation R on a set A is called a partial order, or a partial ordering relation, if R is reflexive, antisymmetric, and transitive.

Definition (7.7) An equivalent relation R on a set A is a relation that is reflexive, symmetric, and transitive.

WEN-CHING LIEN Discrete Mathematics Thank you.

WEN-CHING LIEN Discrete Mathematics