Mathematics Sets and Logic Week 1

Chapter 1 :Introduction to Set and Logic

SM0013 Mathematics I Khadizah Ghazali

Lecture 1 – 25/05/2011

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LEARNING OUTCOMESAt the end of this chapter, you should be able to :

know what set & logic are aboutdefine some basic terminologies in set & logicidentify relations between pairs of setsuse Venn diagrams & the counting formula to solve set equationsbuild & use the truth table

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Outline (A)

1.1 Symbols in set & logic1.2 Set Terminology and Notation1.3 Operations on set1.4 Algebra of sets1.5 Finite Sets and Counting

Principle.

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Outline (B)

1.6 Statements and Argument1.7 Combining, Conditional and

Biconditional Statements1.8 Truth Table1.9 Logic of Equivalent

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SECTION 1.1Symbols in set & logic

Symbols is the central part in set & logic. Here some important symbols :

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∋∋

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Here is a list of sets that we will refer to often, and the symbols are standard.

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Example 1.1Example 1.1

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Example 1.2Example 1.2

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Example 1.3Example 1.3

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SECTION 1.2Set Terminology & Notation

George Cantor (1845-1915), in 1895, was the first to define a set formally.Definition : Set

A set is a group of things of the same kind that belong together.

The objects that make up a set are called

elements or members of the set.

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Some Properties of Sets

The order in which the elements are presented in a set is not important.

A = {a, e, i, o, u} andB = {e, o, u, a, i} both define the same set.

The members of a set can be anything.In a set the same member does not appear more than once.

F = {a, e, i, o, a, u} is incorrect since the element ‘a’ repeats.

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Some Notation

Consider the set A = {a, e, i, o, u} thenWe write “‘a’ is a member of ‘A’” as:

a ∈ A

We write “‘b’ is not a member of ‘A’” as:b ∉ ANote: b ∉ A ≡ ¬ (b ∈ A)

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Set representation:

There are 4 ways to represent a set.

1. Set may be represented by words, for example: A = the first three natural numbers greater

than zeroB = the colors red, white, blue, and green

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Set representation (cont.):

2. Another way to represented a set is to listits elements between curly brackets (by enumeration). ~ is called the roster method, for example: C = {1, 2, 3}D = {red, white, blue, green}

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Set representation (cont.):

3. Another kind of set notation commonly used is set-builder notation, for example:The set E of a natural number less than 4 is written as

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Set representation (cont.):

4. A set can also be represented by a Vann diagram

A pictorial way of representing sets.The universal set is represented by the interior of a rectangle and the other sets are represented by disks lying within the rectangle.

E.g. A = {a, e, i, o, u} a e

i ou

A

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Equality of two SetsA set ‘A’ is equal to a set ‘B’ if and only if both sets have the same elements. If sets ‘A’ and ‘B’are equal we write: A = B. If sets ‘A’ and ‘B’ are not equal we write A ≠ B.In other words we can say:A = B ⇔ (∀x, x∈A ⇔ x∈B)

E.g.A = {1, 2, 3, 4, 5}, B = {2, 4, 1, 3, 5}, C = {1, 3, 5, 4}D = {x : x ∈ Z ∧ 0 < x < 6}, E = {1, 10/5, , 22, 5} then A = B = D = E and A ≠ C.

9

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Universal Set and Empty Set

The members of all the investigated sets in a particular problem usually belongs to some fixed large set. That set is called the universal set and is usually denoted by ‘U’.The set that has no elements is called the empty set and is denoted by Φ or {}.

E.g. {x | x2 = 4 and x is an odd integer} = Φ

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Cardinality of a Set

The number of elements in a set is called the cardinality of a set. Let ‘A’ be any set then its cardinality is denoted by |A| @ n(A)E.g. A = {a, e, i, o, u} then |A| = 5.

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Subsets

Set ‘A’ is called a subset of set ‘B’ if and only if every element of set ‘A’ is also an element of set ‘B’. We also say that ‘A’ is contained in ‘B’ or that ‘B’ contains‘A’. It is denoted by A ⊆ B or B ⊇ A.In other words we can say:(A ⊆ B) ⇔ (∀x, x ∈ A ⇒ x ∈ B)

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Subset cont.

If ‘A’ is not a subset of ‘B’ then it is denoted by A ⊆ B or B ⊇ A

E.g. A = {1, 2, 3, 4, 5} and B = {1, 3} and C = {2, 4, 6} then B ⊆ A and C ⊆ A

1 35

24

6

B A

C

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Some Properties Regarding Subsets

For any set ‘A’, Φ ⊆ A ⊆ UFor any set ‘A’, A ⊆ AA ⊆ B ∧ B ⊆ C ⇒ A ⊆ CA = B ⇔ A ⊆ B ∧ B ⊆ A

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Proper Subsets

Notice that when we say A ⊆ B then it is even possible to be A = B.We say that set ‘A’ is a proper subset of set ‘B’ if and only if A ⊆ B and A ≠ B. We denote it by A ⊂ B or B ⊃ A.In other words we can say:(A ⊂ B) ⇔ (∀x, x∈A ⇒ x∈B ∧ A≠B)

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Venn Diagram for a Proper Subset

Note that if A ⊂ B then the Venn diagram depicting those sets is as follows:

If A ⊆ B then the disc showing ‘B’ may overlap with the disc showing ‘A’ where in this case it is actually A = B

B A

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Power Set

The set of all subsets of a set ‘S’ is called the power set of ‘S’. It is denoted by P(S) or 2S.In other words we can say:P(S) = {x : x ⊆ S}

E.g. A = {1, 2, 3} thenP(A) = {Φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

Note that |P(S)| = 2|S|.E.g. |P(A)| = 2|A| = 23 = 8.

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???

Is the empty set, Φ a subset of {3, 5, 7}?If we say that Φ is not a subset of {3, 5, 7}, then there must be an element of Φ that does not belong to {3, 5, 7}. But that cannot happen because Φ is empty. So Φ is a subset of {3, 5, 7}. In fact, by the same reasoning, the empty set is a subset of every set.

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SECTION 1.3Operations on Set

Complement :The (absolute) complement of a set ‘A’ is the set of elements which belong to the universal set but which do not belong to A. This is denoted by Ac or Ā or Á .In other words we can say:Ac = {x : x∈U ∧ x∉A}

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Venn Diagram for the Complement

A

Ac

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Set Complementation

If U is a universal set and A is a subset of U, then

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∪nion

Union of two sets ‘A’ and ‘B’ is the set of all elements which belong to either ‘A’ or ‘B’ or both. This is denoted by A ∪ B.In other words we can say:A ∪ B = {x : x∈A ∨ x∈B}E.g. A = {3, 5, 7}, B = {2, 3, 5}

A ∪ B = {3, 5, 7, 2, 3, 5} = {2, 3, 5, 7}

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Venn Diagram Representation for Union

BA

A ∪ B

3 57 2

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I∩tersection

Intersection of two sets ‘A’ and ‘B’ is the set of all elements which belong to both ‘A’ and ‘B’. This is denoted by A ∩ B.In other words we can say:A ∩ B = {x : x∈A ∧ x∈B}E.g. A = {3, 5, 7}, B = {2, 3, 5}

A ∩ B = {3, 5}

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Venn Diagram Representation for Intersection

BA

A ∩ B

357 2

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Difference

The difference or the relative complement of a set ‘B’ with respect to a set ‘A’ is the set of elements which belong to ‘A’ but which do not belong to ‘B’. This is denoted by A \ B.In other words we can say:A \ B = {x : x∈A ∧ x∉B}

E.g. A = {3, 5, 7}, B = {2, 3, 5}A \ B = {3, 5, 7}\{2, 3, 5} = {7}

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Venn Diagram Representation for Difference

BA

A B

3 57 2

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Disjoint Sets

If two sets A and B have no elements in common, that is, if

then A and B are called disjoint sets.Notice that the circles corresponding to A and B not overlap anywhere because A∩B is empty.

A ∩ B = Φ

B A

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Some Properties

A ⊆ A ∪ B and B ⊆ A ∪ B A∩B ⊆ A and A∩B ⊆ B|A ∪ B| = |A| + |B| - |A∩B|A ⊆ B ⇒ Bc ⊆ Ac

A B = A ∩ Bc

If A ∩ B = Φ then we say ‘A’ and ‘B’ are disjoint.

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SECTION 1.4Algebra of Set

In section 1.3 we saw that, given two sets A and B, the operations union and intersectioncould be used to generate two further sets

These two new sets can then be combined with a third set C, associated with the same universal set U as the sets A and B, to form four further sets

∪ ∩A B and A B

( ) ( ) ( ) ( )∪ ∪ ∩ ∪ ∪ ∩ ∩ ∩C A B , C A B , C A B , C A B

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And the compositions of these sets are clearly indicated by the shaded portion in the Venn diagrams

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Algebraic Laws on Sets:

Sets operations of union, intersection and complement satisfy various laws (identities).

Let U be the universal set and A, B and C are subsets of U.

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Algebra of Sets

Idempotent lawsA ∪ A = AA ∩ A = A

Associative laws(A ∪ B) ∪ C = A ∪ (B ∪ C)(A ∩ B) ∩ C = A ∩ (B ∩ C)

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Algebra of Sets cont.

Commutative lawsA ∪ B = B ∪ AA ∩ B = B ∩ A

Distributive lawsA ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

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Algebra of Sets cont.

Identity lawsA ∪ Φ = AA ∩ U = AA ∪ U = UA ∩ Φ = Φ

Involution laws(Ac)c = A

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Algebra of Sets cont.

Complement lawsA ∪ Ac = UA ∩ Ac = ΦUc = ΦΦc = U

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Algebra of Sets cont.

De Morgan’s laws(A ∪ B)c = Ac ∩ Bc

(A ∩ B)c = Ac ∪ Bc

Note: Compare these De Morgan’s laws with the De Morgan’s laws that you will find in logic and see the similarity.

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Proofs

Basically there are two approaches in proving above mentioned laws and any other set relationship

Mathematical NotationUsing Venn diagrams

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Example 1.4 :Example 1.4 :

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Solution : a)Method 1;

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Solution : a)Method 2;

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Solution :b)

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SECTION 1.5Finite Sets & Counting Principle

Definition : Finite setA set is said to be finite if it contains exactly p elements. Otherwise a set is said to be infinite.

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Number of the elements in a set;

is determined by simply counting the elements in the set.

If A is any set, then n(A) or |A| denotes the number of elements in A.

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Another result that is easily seen to be true is the following cases :

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Case III : Inclusion-Exclusion Principle

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Case IV:

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Example 1.5 :Example 1.5 :

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Example 1.6 :Example 1.6 :

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Example 1.7 :Example 1.7 :

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