7th topic b.pptx

8/14/2019 7th topic b.pptx

1/15

KS091201 MATEMATIKA DISKRIT W.07.b1

KS091201 MATEMATIKA DISKRIT (DISCRETE MATHEMATICS)

Week 0.7b

Discrete Basic Structure:Set (b)

11/10/2013


2/15

outline

2

Set operations: Union Set operations: Intersection

Set operations: Disjoint

Set operations: Complement

Set operations: Difference Set operations: Complements

Set Identity

Proof By Identity

KS091201 MATEMATIKA DISKRIT W.07.b 11/10/2013


3/15

KS091201 MATEMATIKADISKRIT W.07.b

Recap A set is an unordered collection of objects

aS

N, Z, Z+, Q, R

U: Universal set, = { }

S T if x (x S x T)

S = T if S T and T S.

|A|: the number of elements in a set A.

Power Set: P(S) = {T | T S} A x B = { (a,b) | a A and b B }

11/10/20133


4/15

Refreshment Questions

1. What is the cardinality of :

a) {a}

b) {{a,a}}

c) {a, {a}}d) {a, {a}, {a, {a}} }

2. How many different elements doesA x B

have ifAhas m elements and Bhas n

element

KS091201 MATEMATIKA DISKRIT W.07.b4 11/10/2013


5/15

Refreshment Quiz

1. What is the cardinality of :

a) 1

b) 1

c) 2

d) 3

2. mn elements.

KS091201 MATEMATIKA DISKRIT W.07.b5 11/10/2013


6/15

Set operations: Union

Formal definition for the union of two sets:A UB = {x|xA orxB }

Further examples {1, 2, 3} U{3, 4, 5} = {1, 2, 3, 4, 5}

{a, b} U{3, 4} = {a, b, 3, 4}

{1, 2} U= {1, 2}

Properties of the union operation

A U= A Identity law A UU= U Domination law

A UA = A Idempotent law

A UB = B UA Commutative law

A U(B UC) = (A UB) UC Associative law

6 KS091201 MATEMATIKA DISKRIT W.07.b 11/10/2013


7/15

Set operations: Intersection

Formal definition for the intersection of two sets:A B = {x|xA andxB }

Examples

{1, 2, 3} {3, 4, 5} = {3}

{a, b} {3, 4} =

{1, 2} =

Properties of the intersection operation

A U = A Identity law A = Domination law

A A = A Idempotent law

A B = B A Commutative law

A (B C) = (A B) C Associative law



8/15

Disjoint sets

Formal definition for disjoint sets: two sets are disjoint iftheir intersection is the empty set

Further examples {1, 2, 3} and {3, 4, 5} are not disjoint

{a, b} and {3, 4} are disjoint

{1, 2} and are disjoint Their intersection is the empty set

and are disjoint! Their intersection is the empty set



9/15

Formal definition for the difference of two sets:A - B = {x|xA andxB }

Further examples {1, 2, 3} - {3, 4, 5} = {1, 2}

{a, b} - {3, 4} = {a, b}

{1, 2} - = {1, 2} The difference of any set S with the empty set will be the set S

Set operations: Difference



10/15

Complement sets

Formal definition for the complement of a set:A ={x|xA } = Ac Or UA, where Uis the universal set

Further examples (assuming U= Z) {1, 2, 3}c= { , -2, -1, 0, 4, 5, 6, }

{a, b}c= Z

Properties of complement sets (Ac)c = A Complementation law

A U Ac= U Complement law

A Ac= Complement law



11/15

Set identities

A= AAU = A

Identity Law AU = UA=

Domination law

AA = A

AA = AIdempotent Law (Ac)c= A Complement Law

AB = BAAB = BA

CommutativeLaw

(AB)c= AcBc(AB)c= AcBc

De Morgans Law

A(BC)

= (AB)C

A(BC)= (AB)C

Associative Law

A(BC) =

(AB)(AC)

A(BC) =(AB)(AC)

Distributive Law

A(AB) = A

A(AB) = AAbsorption Law

AAc= U

AAc=Complement Law



12/15

How to prove a set identity

For example: AB=B-(B-A)

Four methods:

Use the basic set identities

Use membership tables

Prove each set is a subset of each other

Use set builder notation and logical equivalences



13/15

What we are going to prove

AB=A-(A-B)

A B

ABA-B



14/15

Proof by Set Identities

AB = A - (A - B)

Proof) A - (A - B) = A - (ABc)

= A(ABc)c

= A(Ac

B) De Morgans Law= (A Ac) (AB) Distributive Law

= (AB) Complement Law

= AB Identity Law



15/15

11/10/2013KS091201 MATEMATIKA DISKRIT W.07.b15

AB = B - (B - A)

Proof) B - (B - A) = B - (B Ac)

= B (B Ac)c

= B (Bc

A)= (B Bc) (B A)

= (B A)

= AB commutative law

7th topic b.pptx

Documents

Transcript of 7th topic b.pptx