discrete mathematics
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Discrete Mathematics. Chapter 1 The Foundations : Logic and Proofs, Sets, and Functions. 大葉大學 資訊工程系 黃鈴玲. 1-1 Logic. Def : A proposition ( 命題 ) is a statement that is either true or false, but not both. Example 1 : The following statements are propositions. - PowerPoint PPT PresentationTRANSCRIPT
Discrete Mathematics
Chapter 1 The Foundations : Logic and Proofs, Sets, and Functions
大葉大學 資訊工程系 黃鈴玲
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1-1 Logic
Def : A proposition ( 命題 ) is a statement that is either true or false, but not both.
Example 1 : The following statements are propositions.
(1) Toronto is the capital of Canada. (F)
(2) 1 + 1 = 2 (T) Example 2 : Consider the following sentences.
(1) what time is it ? (not statement)
(2) Read this carefully. (not statement)
(3) x + 1 = 2 (neither true nor false)
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Logical operators ( 邏輯運算子 ) and truth table ( 真值表 )
Table 1. The truth table for the Negation (not) of a Proposition
eg. p : “ Today is Friday.”
﹁ p : “ Today is not Friday.”
Def : A truth table displays the relationships between the truth values of propositions.
Table 2. The truth table for the Conjunction (and) of two propositions.
eg. p : “ Today is Friday.”
q : “ It’s raining today. ”
p q : “ Today is Friday
and it’s raining
today. “
p q p q
T T T
T F F
F T F
F F F
p ﹁ p
T F
F T
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Table 3. The truth table for the Disjunction (or) of two propositions.
eg. p : “ Today is Friday. “
q : “ It’s raining today . “
p q : “ Today is Friday or
it’s raini
ng today. “ Table 4. The truth table for the Exclusive or (xor) of two prop
ositions.
p q p qT T T
T F T
F T T
F F F
p q p q ⊕T T F
T F T
F T T
F F F
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Table 5. The truth table for the Implication (p implies q) p → q .
( 觀念 : 若 p 對,則 q 一定要對 若 p 錯,則對 q 不做要求 )
eg. p : “ You make more than $25000 ”
q : “ You must file a tax return. “ p → q : “ If you make more … then you must … . “
Some of the more common ways of expressing this implication are : (1) if p then q ( 若 p 則 q , p 是 q 的充分條件 ) (2) p implies q (3) p only if q ( 只有 q 是 True 時, p 才可能是 True ,
若 q 是 False ,則 p 一定是 False)
p q p → q
T T T
T F F
F T T
F F T
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Def : In the implication p → q , p is called the hypothesis ( 假設 )and q is called the conclusion ( 結論 ).
Def : Compound propositions ( 合成命題 ) are formed from existing propositions using logical operators. ( 即 、 、 ⊕、 →等 )
Table 6. The truth table for the Biconditional p ↔ q ( p → q and q → p )
“ p if and only if q “
“ p iff q “ “ If p then q , and
conversely.”
p q p → q q → p p ↔ q
T T T T T
T F F T F
F T T F F
F F T T T
( 若且唯若 )
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Example 9 : How can the following English sentence be translated into a logical expression ?
“ You can access the Internet from campus only if
you are a computer science major or you are not
a freshman. ” Sol :
p : “ You can access the Internet from campus. “
q : “ You are a computer science major. “
r : “ You are a freshman. “
∴ p only if ( q or ( ﹁ r ))
=> p → ( q ( ﹁ r ))
Translating English Sentences into Logical Expression
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Example 10 : You cannot ride the roller coaster ( 雲霄飛車 ) if you are under 4 feet tall unless you are older than 16 years old.
Sol : q : “ You can ride the roller coaster. “
r : “ You are under 4 feet tall. “
s : “ You are older than 16 years old. “
∴ ﹁ q if r unless s
∴ ( r ﹁ s ) → ﹁ q Table 7. Precedence of Logical Operators
eg. (1) p q r means ( p q ) r (2) p q → r means ( p q ) → r
(3) p ﹁ q means p ( ﹁ q )
Exercise : 9 、 13 、 25 、 27 、30
Operator Precedence
﹁ 1
2
3
→ 4
5
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1-2 Propositional Equivalences Def : A compound proposition that is always true
is called a tautology. ( 真理 )
A compound proposition that is always false
is called a contradiction. ( 矛盾 ) Example 1 :
Def : The propositions p and q that have the same truth values in all possible cases are called logically equivalent. The notation p ≡ q ( or p q ) denotes that p and q are logically equivalent.
p ﹁p
p ﹁ p p ﹁ p
T F T F
F T T F
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Example 2 : Show that ﹁ ( p q ) ≡ ﹁ p ﹁ q
pf :
※ Some important logically equivalences (Table 5) (1) p q ≡ q p (2) p q ≡ q p (3) ( p q ) r ≡ p (q r ) (4) ( p q ) r ≡ p (q r ) (5) p ( q r ) ≡ ( p q ) ( p r ) (6) p ( q r ) ≡ ( p q ) ( p r ) ((5) 、 (6) 的觀念類似於 (a + b) x c = (a x c ) + (b x c))
p q ﹁ ( p q ) ﹁ p ﹁ q ﹁ p ﹁ q
T T F F F F
T F F F T F
F T F T F F
F F T T T T
commutative laws. 交換律
associative laws. 結合律
distributive laws 分配律
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(7) ﹁ ( p q ) ≡ ﹁ p ﹁ q (8) ﹁ ( p q ) ≡ ﹁ p ﹁ q (9) p ﹁ p ≡ T (10) p ﹁ p ≡ F (11) p → q ≡ ﹁ p q
Example 5 : Show that ﹁ ( p ( ﹁ p q )) ≡ ﹁ p ﹁ q pf : ( 也可用真值表証 ) ﹁ ( p ( ﹁ p q ) ) ≡ ﹁ p ﹁ ( ﹁ p q )
≡ ﹁ p ( p ﹁ q ) ≡ ( ﹁ p p ) ( ﹁ p ﹁ q ) ≡ F ( ﹁ p ﹁ q ) ≡ ﹁ p ﹁ q
De Morgan’s laws
by (8)
by (7)
by (6)
by (10)
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Example 6 : Show ( p q ) → (p q) is a tautology. pf : ( p q ) → (p q) ≡ ﹁ ( p q ) (p q )
≡ ( ﹁ p ﹁ q ) (p q )
≡ ( ﹁ p p ) ( ﹁ q q )
≡ T T
≡ T
Exercise : 7 、 9 、 17
By (3)
By (11)
By (7)
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1-3 Predicates and Quantifiers
目標 : 了解 “ ∀ “ 及 “ ∃ “ 符號 Def : The statement P(x) is said to be the value of the
propositional function P at x . ex :
P(x) : “ x is greater than 3 “
※ 命題中出現變數 x 時 the universe of discourse (or domain) of x
指的是 x 的範圍 ※Quantifiers : ( 數量詞,如 some , any , all 等 )
∀ : universal quantifier ( for all ) ∃ : existential quantifier ( there exist , there is , for some
)
variable predicate
屬性 數量詞
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Table 1. Quantifiers
Example 13 : Let P(x) : x2 > 10, when x , x ∈ ≤ 4What is the truth value of x P(x) ∃ ?
Sol : x ∈ {1, 2, 3, 4} ∴ 42 = 16 > 10
∴ ∃x P(x) is true.
Statement When True ? When False ?
∀x P(x) P(x) is true for every x.
There is an x for which P(x) is false.
∃x P(x) There is an x for which P(x) is true.
P(x) is false for every x.
+
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Table 2. Negating Quantifiers.
Example 16 : P(x) : x2 > x , Q(x) : x2 = 2 , what is the negations of x P(x) and x Q(x) ?∀ ∃
Sol : ﹁∀ x P(x) ≡ x ∃ ﹁ P(x) ≡ x (x∃ 2 ≤ x)
﹁∃ x Q(x) ≡ x ∀ ﹁ Q(x) ≡ x (x∀ 2 ≠ x)
Exercise : 11 、 13 、 15 、 49
Negation Equivalent Statement
When True ? When False ?
﹁∃ x P(x)
∀x ﹁ P(x) P(x) is false for every x.
∃x, s.t. P(x) is true.
﹁∀ x P(x)
∃x ﹁ P(x) ∃x, s.t. P(x) is false
P(x) is true for every x.
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補充 : 習題 48 “ ∃! ” 表示 “ 存在且唯一 “ ∃!x P(x) 表示 “ There exists a unique x s.t. P(x) i
s true. ” Example : What is the truth values of the state
ments (a) ! x ( x∃ 2 = 1 ) (b) ! x ( x + 3 = 2x )∃
where the universe of discourse is the set of integers. ( 即 x )∈
Ans : (a) 12 = 1 , (-1)2=1 (b) True.
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1-4 Nested Quantifiers
eg. x y (x + y = 0 ) ∀ ∃ Table 2. Quantifications of Two Variables.
Statement When true ? When False ?
∀x y P(x,y)∀∀y x P(x,y)∀
P(x,y) is true for every pair x,y. ①
∃a pair (x,y) s.t. P(x,y) is false. ③
∀x y P(x,y)∃ For every x , y s.t. P(x,y) i∃s true. ②
∃x , s.t. P(x,y) is false for every y. ④
∃x y P(x,y)∀ There is an x for which P(x,y) is true for every y. ③
For every x, y s.t. P(x,y) ∃is false. ②
∃x y P(x,y)∃∃y x P(x,y)∃
∃ a pair (x,y) s.t. P(x,y) is true. ③
∀ pair (x,y) , P(x,y) is false. ⑤
例 : p(x,y) : x + y ① ≥0 , x,y N p(x,y) : xy = 0 , x,y Z∈ ③ ∈ ② p(x,y) : x + y = 2 , x,y Z p(x,y) : xy = -1 , x,y Z∈ ④ ∈ ⑤ p(x,y) : x + y = ½ , x,y Z∈ Exercise: 27
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1-6 Sets
Def 1 : A set is an unordered collection of objects. Def 2 : The objects in a set are called the elements , or m
embers of the set. Example 4 : 常見的重要集合
N = { 0,1,2,3,…} , the set of natural number ( 自然數 ) Z = { …,-2,-1,0,1,2,…} , the set of integers ( 整數 ) Z+ = { 1,2,3,…} , the set of positive integers ( 正整數 ) Q = { p / q | p Z , q Z , q≠0 } , the set of ∈ ∈ rational num
bers ( 有理數 ) R = the set of real numbers ( 實數 )
( 元素可表示成 1.234 等小數形式 )
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Def 4 : A ⊆ B iff x , x ∀ A ∈ → x B ∈補充: A ⊂ B 表示 A ⊆ B 但 A ≠ B
Def 5 : S : a finite set
The cardinality of S , denoted by |S| , is the number of elements in S.
Def 7 : S : a set
The power set of S , denoted by P(S) , is the set of all subsets of S.
Example 11 : S = {0,1,2}
P(S) = {, {0} , {1} , {2} , {0,1} , {0,2} , {1,2} , {0,1,2} } Def : A , B : sets The Cartesian Product of A and B ,
denoted by A x B , is the set A x B = { (a,b) | a A and b B }∈ ∈
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Note. |A x B| = |A| . |B| Example 14 :
A = {1,2} , B = {a, b, c}
A x B = {(1,a) , (1,b) , (1,c) , (2,a) , (2,b) , (2,c)}
Exercise : 5 、 7 、 8 、 13 、 17 、 19
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1-7 Set Operations
Def 1,2,4 : A,B : sets A B = { x | x ∪ A or x B } (union) A∩B = { x | x A and x B } (intersection) A – B = { x | x A and x B } ( 也常寫成 A \
B) Def 3 : Two sets A,B are disjoint if A∩B = Def 5 : Let U be the universal set. The complement of the set A , denoted by A , is
the set U – A . Example 10 : Prove that A∩B = A B ∪ pf :
稱為 Venn Diagram
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Def 6 : A1 , A2 , … , An : sets
Let I = {1,3,5} ,
Def : (p.95 右邊 ) A,B : sets
The symmetric difference of A and B , denoted by A B⊕ , is the set
{ x | x A B or x B A } = ( A B ) ∪ ( A ∩B ) ※Inclusion – Exclusion Principle ( 排容原理 )
|A ∪ B| = |A| + |B| |A ∩ B| Exercise : 10,37
n
n
iAAAA 21
1
n
n
iAAAA 21
1
531 AAAAIi
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1-8 Functions
Def 1 : A,B : sets
A function f : A → B is an assignment of exactly one element of B to each element of A. We write f(a) = b if b is the unique element of B assigned by f to a A. ∈
eg. A B A B
1
2
3
1
2
α
β
γ
α
β
γ
Not a function Not a function
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Def : ( 以 f : A→B 為例,右上圖 )
f (α) = 1 , f (β) = 4 , f (γ) = 2
1 稱為 α 的 image ( 必唯一 ) , α 稱為 1 的 pre-image( 可能不唯一 )
A : domain of f , B : codomain of f
range of f = {f (a) | a A} = ∈ f (A) = {1,2,4} ( 未必 =B)
Example 2 : f : Z → Z , f (x) = x2 , 則 f 的 domain , codomain
及 range ?
A B
1
2
α
β
γ
A B
1
2
3
α
β
γ4
a function a function
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Example 4 : Let f1 : R → R and f2 : R → R s.t.
f1(x) = x2 , f2(x) = x - x2 , What are the function f1 + f2 and f1 f2 ?
Sol :
( f1 + f2 )(x) = f1(x) + f2(x) = x2 + ( x – x2 ) = x
(f1 f2)(x) = f1(x) . f2(x) = x2( x – x2 ) = x3 – x4
Def : A function f is said to be one-to-one , or injective , iff f (x) ≠ f (y) whenever x ≠ y.
Example 6 : A B
12
a
b
c
A B12
a
b
c
d
45
3d
34
5
是 1-1 不是 1-1 , 因 g(a) = g(d) = 4
f g
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Example 8 : Determine whether the function f (x) = x + 1 is one-to-one ?
Sol : x ≠ y x + 1 ≠ y + 1
f (x) ≠ f (y)
∴ f is 1-1 Def 7 : A function f : A → B is called onto , or surjective , iff
for every element b B , ∈ a ∃ A with ∈ f (a) = b. ( 即 B 的所有元素都被 f 對應到 )
Example 9 :
Note : 當 |A| < |B| 時,必定不會 onto.
noto
a
bcd
2
3
1
f
not noto
A B
a
b
c
1234
f
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Def 8 : The function f is a one-to-one correspondence , or a bijection , if it is both 1-1 and onto.
Examples in Fig 5
※ 補充 : f : A →B (1) If f is 1-1 , then |A| ≤ |B| (2) If f is onto , then |A| ≥ |B| (3) if f is 1-1 and onto , then |A| = |B|.
1-1 , onto
a
b
c
2
3
1
4
not 1-1 , onto
ab
c
1
2
3d
1-1 and onto
a
bcd
23
1
4
28
※Some important functions Def 12 :
floor function : x : ≤ x 的最大整數,即 [ x ] ceiling function : x : ≥ x 的最小整數 .
Example 21 : ½ = -½ = 7 = ½ = -½ = 7 =
Example 26 : factorial function :
f : N → Z+ , f (n) = n! = 1 x 2 x … x n Exercise : 1,12,17,19