Ch1 sets and_logic(1)

Chapter 1 Sets and Logic 2008 학년도 2 학기 고려대학교 과학기술대학 컴퓨터 정보학과

1.1 Sets ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Symbol Set R Set of all REAL numbers Z Set of all INTEGERs Q Set of all RATIONAL numbers superscript indicate ＋ Positive － Negative nonneg nonnegative

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Cardinality &Empty Set A = {1, 2, 3, 4} , |A| = ? |{R, Z}| = ?

[object Object],[object Object],[object Object],Equal Two sets X and Y are equal if X and Y have the same elements. X = Y if for every x , if x ∈ X, then x ∈ Y and for every y , if y ∈ Y, then y ∈ X A = {1, 3, 2}, B = {2, 3, 2, 1}. A = B ? Let us prove that if A = { x | x 2 + x – 6 = 0 } and B = {2, -3} Then A = B .

Subset ,[object Object],If A and B are sets, then A is called a subset of B , written A⊆ B , If, and only if, every element of A is also an element of B . Symbolically : A ⊆ B ⇔ ∀ x , if x ∈ A then x ∈ B. The phrases A is contained in B and B contains A are alternative ways of saying that A is a subset of B. A set A is not a subset of a set B , written A ⊆ B, if, and only if, there is at least one element of A that is not an element of B . Symbolically : ,[object Object]

[object Object],[object Object],[object Object],Subset Z ⊆ Q ? Let X = { x | 3 x 2 – x – 2 = 0 }. X ⊆ Z ? C = {1, 3}, B = {1, 2, 3, 4}. C ⊆ A ?

Proper Subset ,[object Object],[object Object],[object Object],Let A and B be sets. A is a proper subset of B , if, and only if, every element of A is in B but there is at least one element of B that is not A . A B A = B A B A B B A A B A = B A ⊆ B A ⊆ B A⊂ B

Power Set ,[object Object],[object Object],Given a set A , the power set of A , denoted P(A), is set of all subsets of A Find the power set of the set A ={ a, b, c }. That is, find P ( A) . P ( A ) = ,[object Object],For all integers n ≥ 0, if a set X has n elements, then P(X ) has 2 n elements

Set Operations ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],A ={1, 3, 5}, B = {4, 5, 6} A ∪ B = A ∩ B = A – B = B – A = Since Q ⊆ R, R ∪ Q = R ∩ Q = R – Q = Q – R = ,[object Object]

Disjoint ,[object Object],[object Object],Two sets are called disjoint if, and only if, they have no elements in common. Symbolically : A and B are disjoint ⇔ A ∩ B = Φ - {1, 4, 5} and {2, 6} - S = {{1, 4, 5,}, {2, 6}, {3}, {7, 8}}

Set Operations ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Set Identity ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Set Identity ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],For i ≥ 1, define A i = { i, i+1 , … } and S = { A 1 , A 2 , …, A n }. Then, and ?

Partition of sets ,[object Object],[object Object],[object Object],Sets A 1 , A 2 , A 3 , …, A n are mutually disjoint if, and only if, no two sets A i and A j with distinct subscripts have any elements in common. More precisely, for all i, j = 1, 2, 3, …, n, A i ∩ A j = Φ whenever i ≠ j ,[object Object],[object Object],[object Object],A 1 A 2 A 3 … A n

Cartesian product ,[object Object],[object Object],[object Object],[object Object],Let n be a positive integer and let x 1 , x 2 , x 3 , … x n be elements. The ordered n-tuple , ( x 1 , x 2 , x 3 , … x n ), consists of x 1 , x 2 , x 3 , …, x n together with the ordering. An ordered 2-tuple is called an ordered pair ; and an ordered 3-tuples is called an ordered triple . Two ordered n-tuples ( x 1 , x 2 , x 3 , …, x n ) and ( y 1 , y 2 , y 3 , …, y n ) are equal if, and only if, x 1 = y 1 , x 2 = y 2 , x 3 = y 3 , …, x n = y n . Symbolically : ( x 1 , x 2 , x 3 , …, x n ) = ( y 1 , y 2 , y 3 , …, y n ) ⇔ x 1 = y 1 , x 2 = y 2 , x 3 = y 3 , …, x n = y n

Cartesian product ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Given two sets A and B , the Cartesian Product of A and B , denote A × B ( read “ A cross B ” ), is the set of all ordered pairs ( a, b ), where a is in A and b is in B . Given A 1 × A 2 × A 3 , …, × A n , is the set of all ordered n-tuples ( a 1 , a 2 , a 3 , …, a n ) where a 1 ∈ A 1 , a 2 ∈ A 2 , a 3 ∈ A 3 , …, a n ∈ A n . Symbolically : A × B = { ( a, b ) | a ∈ A and b ∈ B }, A 1 × A 2 × A 3 ×… × A n = { ( a 1 , a 2 , a 3 , …, a n ) | a 1 ∈ A 1 , a 2 ∈ A 2 , a 3 ∈ A 3 , …, a n ∈ A n }.

Logic ,[object Object],[object Object],[object Object],[object Object]

1.2 Propositions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],A proposition ( 명제 ) is a statement or sentence that can be determined to be either true or false but not both

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Propositions

Connectives ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Conjunction(  ) ,[object Object],[object Object],If p : It is raining q : It is cold then what is the conjunction of p and q ? p q p  q T T T T F F F T F F F F

[object Object],[object Object],If p : A decade is 10 years q : A millennium is 100 years then what is the conjunction of p and q ? Is it true or false? x < 10 && y > 4

Disjunction(  ) ,[object Object],[object Object],[object Object],If p : A millennium is 100 years q: A millennium is 1000 years then what is the disjunction of p and q ? Is it true or false? x < 10 || y > 4 p q p v q T T T T F T F T T F F F

Exclusive OR ,[object Object],[object Object],p q p ∨ q p ∧ q ~(p ∧ q) ( p ∨ q ) ∧ ~ ( p ∧ q ) T T T T F F T F T F T T F T T F T T F F F F T F

Logical connectives in C programs [Ex] int i, j; i = 2 && ( j = 2 ); printf(“%d %d”, i, j); /* 1 2 is printed */ ( i = 0 ) && ( j = 3 ); printf(“%d %d”, i, j); /* 0 2 is printed */ i = 0 || ( j = 4 ); printf(“%d %d”, i, j); /* 1 4 is printed */ ( i = 2 ) || ( j = 5 ); printf(“%d %d”, i, j); /* 2 4 is printed */

Negation(  ) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object], ! ( x < 10) p  p T F F T

More compound statements ,[object Object],[object Object],[object Object],p q r p ∧ q  r ( p ∧ q ) ∨  r T T T T F T T T F T T T T F T F F F T F F F T T F T T F F F F T F F T T F F T F F F F F F F T T

1.3 Conditional propositions and logical equivalence ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Implication(  ) ,[object Object],p  q is true when both p and q are true or when p is false true by default or vacuously true p q p  q T T T T F F F T T F F T

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Implication ,[object Object]

Implication ,[object Object],[object Object],[object Object],A necessary condition is expressed by the conclusion. A sufficient condition is expressed by the hypothesis.

Double Implication(  ) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],p q p -> q q -> p ( p -> q )∧( q -> p ) p ↔ q T T T T T T T F F T F F F T T F F F F F T T T T

Logical Equivalence ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Logical Equivalence ,[object Object],Rewrite the following statement in if-then form Either you get to work on time or you are fired. ,[object Object],p q p -> q ~p ~p ∨ q T T T F T T F F F F F T T T T F F T T T

Logical Equivalence ,[object Object],[object Object],[object Object],[object Object],[object Object],If Jerry receives a scholarship, then he goes to college ,[object Object],[object Object],[object Object],[object Object]

Contrapositive ,[object Object],The contrapositive( 대우 ) of p -> q is ~ q -> ~ p If the network is down, then Dale cannot access the Internet ,[object Object],The conditional statement and its contrapositive are logically equivalent ,[object Object],p q p -> q ~q ~p ~q -> ~p T T T F F T T F F T F F F T T F T T F F T T T T

Converse & Inverse ,[object Object],[object Object],[object Object],[object Object],The converse( 역 ) of p -> q is q -> p The inverse( 이 ) of p -> q is ~ p -> ~ q p q p -> q q -> p ~p ~q ~p -> ~q T T T T F F T T F F T F T T F T T F T F F F F T T T T T

Tautology & Contradiction 합성 명제의 진리값이 항상 T 인 명제 , 즉 , 합성명제를 구성하고 있는 단순명제들의 진리값에 상관없이 항상 T 의 진리값을 가진 명제 합성명제의 진리값이 항상 F 인 명제 , 즉 , 합성명제를 구성하고 있는 단순명제들의 진리값에 상관없이 항상 F 의 진리값을 가진 명제 ,[object Object],[object Object]

Tautology & Contradiction ,[object Object],[object Object],[object Object],Contradiction Tautology ,[object Object],p t p ∧ t T T T F T F p c p ∧ c T F F F F F p ~p p ∨ ~p p ∧ ~p T F T F F T T F

Ch1 sets and_logic(1)

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