Logic and Discrete Mathematics. Section 2.5 Equivalence relations and partitions

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1 Logic and Discrete Mathematics Section 2.5 Equivalence relations and partitions Slides version: January 2015

2 Equivalence relations Let X be a set and R X X a binary relation on X. We call R an equivalence relation on X if it is reflexive, symmetric and transitive, i.e., if for all x, y, z X it holds that 1. xrx,

3 Equivalence relations Let X be a set and R X X a binary relation on X. We call R an equivalence relation on X if it is reflexive, symmetric and transitive, i.e., if for all x, y, z X it holds that 1. xrx, 2. if xry then yrx, and

4 Equivalence relations Let X be a set and R X X a binary relation on X. We call R an equivalence relation on X if it is reflexive, symmetric and transitive, i.e., if for all x, y, z X it holds that 1. xrx, 2. if xry then yrx, and 3. if xry and yrz, then xry.

5 Equivalence relations Let X be a set and R X X a binary relation on X. We call R an equivalence relation on X if it is reflexive, symmetric and transitive, i.e., if for all x, y, z X it holds that 1. xrx, 2. if xry then yrx, and 3. if xry and yrz, then xry. Example The following are equivalence relations on the set Z: 1. The equality relation =.

6 Equivalence relations Let X be a set and R X X a binary relation on X. We call R an equivalence relation on X if it is reflexive, symmetric and transitive, i.e., if for all x, y, z X it holds that 1. xrx, 2. if xry then yrx, and 3. if xry and yrz, then xry. Example The following are equivalence relations on the set Z: 1. The equality relation =. 2. The relation of having the same absolute value, denoted abs.

7 Equivalence relations Let X be a set and R X X a binary relation on X. We call R an equivalence relation on X if it is reflexive, symmetric and transitive, i.e., if for all x, y, z X it holds that 1. xrx, 2. if xry then yrx, and 3. if xry and yrz, then xry. Example The following are equivalence relations on the set Z: 1. The equality relation =. 2. The relation of having the same absolute value, denoted abs. 3. The relation of having the same remainder when divided by 3, denoted 3.

8 Equivalence classes Let R X X be an equivalence relation on X and x X. The subset [x] R = {y X xry} is called the equivalence class (or the cluster) of x generated under R.

9 Equivalence classes Let R X X be an equivalence relation on X and x X. The subset [x] R = {y X xry} is called the equivalence class (or the cluster) of x generated under R. Proposition For every equivalence relation R on a set X and x, y X the following hold: 1. x [x] R,

10 Equivalence classes Let R X X be an equivalence relation on X and x X. The subset [x] R = {y X xry} is called the equivalence class (or the cluster) of x generated under R. Proposition For every equivalence relation R on a set X and x, y X the following hold: 1. x [x] R, 2. x [y] R implies [x] R = [y] R,

11 Equivalence classes Let R X X be an equivalence relation on X and x X. The subset [x] R = {y X xry} is called the equivalence class (or the cluster) of x generated under R. Proposition For every equivalence relation R on a set X and x, y X the following hold: 1. x [x] R, 2. x [y] R implies [x] R = [y] R, 3. x [y] R implies [x] R [y] R =,

12 Equivalence classes Let R X X be an equivalence relation on X and x X. The subset [x] R = {y X xry} is called the equivalence class (or the cluster) of x generated under R. Proposition For every equivalence relation R on a set X and x, y X the following hold: 1. x [x] R, 2. x [y] R implies [x] R = [y] R, 3. x [y] R implies [x] R [y] R =, 4. if [x] R [y] R then [x] R [y] R =.

13 Equivalence classes Examples 1. [3] = = {3} [0] = = {0} [ 99] = = { 99}

14 Equivalence classes Examples 1. [3] = = {3} [0] = = {0} [ 99] = = { 99} 2. [2] abs = { 2, 2} [ 50] abs = { 50, 50} [0] abs = {0}

15 Equivalence classes Examples 1. [3] = = {3} [0] = = {0} [ 99] = = { 99} 2. [2] abs = { 2, 2} [ 50] abs = { 50, 50} [0] abs = {0} 3. [4] 3 = {..., 5, 2, 1, 4, 7, 10,...}.

16 Quotient sets Let R X X be an equivalence relation. The set X/ R = {[x] R x X}, consisting of all equivalence classes of elements of X under R, is called the quotient set of X by R.

17 Quotient sets Let R X X be an equivalence relation. The set X/ R = {[x] R x X}, consisting of all equivalence classes of elements of X under R, is called the quotient set of X by R. The equivalence class [x] R, also denoted by x/ R, is also called the quotient-element of x by R.

18 Quotient sets Let R X X be an equivalence relation. The set X/ R = {[x] R x X}, consisting of all equivalence classes of elements of X under R, is called the quotient set of X by R. The equivalence class [x] R, also denoted by x/ R, is also called the quotient-element of x by R. The mapping η R : X X R defined by η R (x) = [x] R is called the canonical mapping of X onto X/ R.

19 Example The quotient set Z/

20 Partitions A partition of a set X is any family P of non-empty and pairwise disjoint subsets of X, the union of which is X.

21 Partitions A partition of a set X is any family P of non-empty and pairwise disjoint subsets of X, the union of which is X. Example The family is a partition of the set {{d, b, c}, {a}, {g, f }, {e, h}} {a, b, c, d, e, f, g, h}.

22 Equivalence relation and partitions Proposition If R is an equivalence relation on a set X, then the quotient set is a partition of X. {[x] R x X}

23 Equivalence relation and partitions Proposition If R is an equivalence relation on a set X, then the quotient set is a partition of X. {[x] R x X} Proposition If P is a partition of a set X then the relation P X X defined by x P y iff x and y belong to the same member of P is an equivalence relation on X.

24 Equivalence relation and partitions examples (1) The quotient set Z/ abs of Z by abs is a partition of Z into infinitely many sets, namely {0}, { 1, 1}, { 2, 2}, { 3, 3},....

25 Equivalence relation and partitions examples (1) The quotient set Z/ abs of Z by abs is a partition of Z into infinitely many sets, namely {0}, { 1, 1}, { 2, 2}, { 3, 3},.... For each integer n the canonical mapping η abs is given by: η abs (n) = { n, n}.

26 Equivalence relation and partitions examples (2) Describe the equivalence relation P corresponding to the partition P of the set {1, 2, 3,..., 12} pictured:

27 The kernel equivalence of a mapping Let f : X Y. Then the kernel equivalence of f is the relation f X X defined such that x f y iff f (x) = f (y).

28 The kernel equivalence of a mapping Let f : X Y. Then the kernel equivalence of f is the relation f X X defined such that x f y iff f (x) = f (y). Theorem Let f : X Y. Then the kernel equivalence of f is an equivalence relation on X.

29 The kernel equivalence of a mapping Let f : X Y. Then the kernel equivalence of f is the relation f X X defined such that x f y iff f (x) = f (y). Theorem Let f : X Y. Then the kernel equivalence of f is an equivalence relation on X. Example Let f : Z Z be defined by f (x) x. Describe f.

30 The mapping µ Theorem Let f : X Y, and let µ : X/ f Y be the mapping defined as µ([x] f ) = f (x).

31 The mapping µ Theorem Let f : X Y, and let µ : X/ f Y be the mapping defined as µ([x] f ) = f (x). Then 1. µ is well defined,

32 The mapping µ Theorem Let f : X Y, and let µ : X/ f Y be the mapping defined as µ([x] f ) = f (x). Then 1. µ is well defined, 2. µ is an injection and

33 The mapping µ Theorem Let f : X Y, and let µ : X/ f Y be the mapping defined as µ([x] f ) = f (x). Then 1. µ is well defined, 2. µ is an injection and 3. f = µη f.

1. Relations 2. Equivalence relations 3. Modular arithmetics. ! Think of relations as directed graphs! xry means there in an edge x!

1. Relations 2. Equivalence relations 3. Modular arithmetics. ! Think of relations as directed graphs! xry means there in an edge x! 2 Today s Topics: CSE 20: Discrete Mathematics for Computer Science Prof. Miles Jones 1. Relations 2. 3. Modular arithmetics 3 4 Relations are graphs! Think of relations as directed graphs! xry means there