Binary Relations Exercises Week 2 Relations Discrete Math Marie Demlová http://math.feld.cvut.cz/demlova February 27, 2020 Marie Demlova: Discrete Math
Operations with Relations Binary Relations Relations on a Set Exercises Equivalence Relations Binary Relations A relation (more precisely a binary relation) from a set A into a set B is any set of ordered pairs R ⊆ A × B . If A = B we speak about a relation on a set A . Examples. ◮ To be a subset. Objects are subsets of a given set U ; a subset X is related to a subset Y if X is a subset of Y . ◮ To be greater or equal. Objects are numbers; a number n is related to a number m if n is greater than or equal to m . ◮ To be a student of a study group. Objects are first year students and study groups; a student a is related to a study group number K if student a belongs to study group K . ◮ The sine function. Consider real numbers; a number x is related to a number y if y = sin x . Marie Demlova: Discrete Math
Operations with Relations Binary Relations Relations on a Set Exercises Equivalence Relations Operations with Relations Set operations Let R and S be two relations from a set A into a set B . ◮ The intersection of relations R and S is R ∩ S ; ◮ The union of R and S is R ∪ S ; ◮ The complement of R is R = ( A × B ) \ R . Inverse Relation. Given R a relation from A into B . Then the inverse relation of the relation R is R − 1 from B into A , defined x R − 1 y if and only if y R x . Marie Demlova: Discrete Math
Operations with Relations Binary Relations Relations on a Set Exercises Equivalence Relations Operations with Relations Composition of Relations. Given R a relation from A into B and S a relation from B into C . Then the composition R ◦ S (sometimes also called the product ), is the relation from A into C defined by: a ( R ◦ S ) c iff there is b ∈ B such that a R b and b S c . Proposition. The composition of relations is associative. I.e., if R is a relation from A to B , S is a relation from B to C , and T is a relation from C to D then R ◦ ( S ◦ T ) = ( R ◦ S ) ◦ T . Marie Demlova: Discrete Math
Operations with Relations Binary Relations Relations on a Set Exercises Equivalence Relations Operations with Relations Proposition. The composition of relations is not commutative. It is not the case that R ◦ S = S ◦ R holds for all relations R and S . Example. Let A be the set of all people in the Czech Republic. Consider the following two relations R , S defined on A : a R b iff a is a sibling of b and a � = b . c S d iff c is a child of d . Then R ◦ S � = S ◦ R . Marie Demlova: Discrete Math
Operations with Relations Binary Relations Relations on a Set Exercises Equivalence Relations Relations on a Set Properties of relations on a set. We say that relation R on A is ◮ reflexive if for every a ∈ A it is a R a ; ◮ symmetric if for every a , b ∈ A it holds that: a R b implies b R a ; ◮ antisymmetric if for every a , b ∈ A it holds that: a R b and b R a imply a = b ; ◮ transitive if for every a , b , c ∈ A it holds that: if a R b and b R c then a R c . Marie Demlova: Discrete Math
Operations with Relations Binary Relations Relations on a Set Exercises Equivalence Relations Equivalence Relations A relation R on A is equivalence if it is reflexive, symmetric and transitive. Given an equivalence relation R on A . An equivalence class of R corresponding to a ∈ A is the set R [ a ] = { b ∈ A | a R b } . Example 1. Then relation R is an equivalence on Z : if and only if m − n is divisible by 12 , ( m , n ∈ Z ) . m R n For R from Example 1 there are twelve distinct equivalence classes, namely R [ i ] , i = 0 , 1 , . . . , 11. Marie Demlova: Discrete Math
Operations with Relations Binary Relations Relations on a Set Exercises Equivalence Relations Equivalence Relations Properties of the Set of Equivalence Classes. Let R be an equivalence on A . The set { R [ a ] | a ∈ A } has the following properties: ◮ Every a ∈ A belongs to R [ a ] ; so � { R [ a ] | a ∈ A } = A . ◮ Equivalence classes R [ a ] are pairwise disjoint. That is, if R [ a ] ∩ R [ b ] � = ∅ , then R [ a ] = R [ b ] . Partition. Let A be a non-empty set. A set S of non-empty subsets of A is a partition of A if the following hold: 1. Every a ∈ A belongs to some member of S , i.e. � S = A . 2. The sets in S are pairwise disjoint. I.e., if X ∩ Y � = ∅ then X = Y for all X , Y ∈ S . Marie Demlova: Discrete Math
Operations with Relations Binary Relations Relations on a Set Exercises Equivalence Relations Equivalence Relations Proposition. Let S be a partition of A . Then the relation R S defined by: a , b ∈ X for some X ∈ S a R S b if and only if is an equivalence on A . If we start with an equivalence R , form the corresponding partition into classes of R , and finally we make the equivalence relation corresponding to the partition, we get the equivalence R . If we start with a partition, then form corresponding equivalence, and finish with the partition into classes of the equivalence, we get the original partition. Marie Demlova: Discrete Math
Binary Relations Exercises Exercises Exercise 1. Write the following relations on a set A as sets of ordered pairs: a) A is the set of all subsets of the set { 1 , 2 } , relation R is “to be a proper subset”. This means that for X , Y ∈ A we have X R Y if and only if X ⊆ Y and X � = Y . b) A = { 2 , 4 , 5 , 8 , 45 , 60 } , R is the relation of divisibility; i.e. m R n if and only if m divides n . Marie Demlova: Discrete Math
Binary Relations Exercises Exercises Exercise 2. A relation R on a closed interval A = [ 0 , 4 ] is given by: x 2 + y 2 + 7 ≤ 4 x + 4 y . if and only if x R y Decide a) whether 2 ( R ◦ R ) 2 and b) whether 0 ( R − 1 ◦ R ) 3. Exercise 3. A relation R on a closed interval A = [ 0 , 1 ] is given by: x R y if and only if y = 2 | x − 1 2 | . Sketch in a plane (as a set of ordered pairs) the relations R , R − 1 and R ◦ R − 1 . Marie Demlova: Discrete Math
Binary Relations Exercises Exercises Exercise 4. Give the properties of the following relations on the set of all natural numbers N : a) m R n if and only if m divides n ; b) m R n if and only if m + n ≥ 50; c) m R n if and only if m + n is even; d) m R n if and only if m · n is even; e) m R n if and only if m = n k for some k ∈ N ; f) m R n if and only if m + n is a multiple of 3; g) m R n if and only if m > n . Marie Demlova: Discrete Math
Binary Relations Exercises Exercises Exercise 5. In the following examples S is a relation on a set A and x , y are elements of set A . Decide whether S is reflexive, symmetric, antisymmetric, transitive. Is it an equivalence, an order relation? a) A is the set of all complex numbers, x S y iff | x | = | y | . b) A is the set of all complex numbers, x S y iff | x | < | y | . c) A is the set of all real numbers, x S y iff x − y is a rational number. d) A is the set of all triangles of a given plane, two triangles are related in S iff they are congruent. e) A is the set of all triangles of a given plane, two triangles are related in S iff they are similar. f) A is the set of all subsets of a set B , two subsets X , Y of the set B are related in S iff they have the same cardinality; i.e., iff there exists an injective mapping of X onto Y . Marie Demlova: Discrete Math
Binary Relations Exercises Exercises Exercise 6. Given two relations R and S from a set A into a set B . Decide whether the following is true: a) ( R ∪ S ) − 1 = R − 1 ∪ S − 1 ; b) ( R ∩ S ) − 1 = R − 1 ∩ S − 1 . Marie Demlova: Discrete Math
Binary Relations Exercises Exercises Exercise 7. Given two relations R and S on a set A . Decide whether it is true: a) If R and S are reflexive, then so is R ◦ S . b) If R and S are symmetric, then so is R ◦ S . c) If R and S are antisymmetric, then so is R ◦ S . d) If R and S are transitive, then so is R ◦ S . Marie Demlova: Discrete Math
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