Chapter 2 Symbolic Logic. Section 2-1 Truth, Equivalence and Implication.
-
Upload
jessica-chandler -
Category
Documents
-
view
227 -
download
4
Transcript of Chapter 2 Symbolic Logic. Section 2-1 Truth, Equivalence and Implication.
Chapter 2
Symbolic Logic
Section 2-1
Truth, Equivalence and Implication
More on Implication
Universal Implication: A statement p implies a statement q, if q is true in every situation that makes p true. pq
EX: (x>2) (x>1)
More on Implication
Ex:
Show that p ^ q implies p V (¬p ^ q)
Def.
A universally true statement is true for each element of the universe.
Ex:
Universe: Flipping 2 coins.
p: if there is one tail then there is one head.
- p is a universally true statement.
Tautology
A tautologically true statement is a statement that is always true ( it can be written as a symbolic statement whose truth table has only Trues in the final column).
Ex: “The result has 2 H’s or the result doesn’t have 2 H’s” ( p V ¬p)
The statement is always true (tautology).
Contradiction
A statement that is always false.
Ex: p ^ ¬ p is always false ( a contradiction).
Example
p is the statement: x<=0 q is the statement: x>=10
Show that ¬(p V q) and ¬p ^ ¬q are equivalent.
Section 2.3
Predicate Logic
Propositional Logic
In propositional logic we used symbols to represent simple statements ( p, q, r, s)
we also used symbols and logical connectives ( V, ^, , , ¬ ) to represent ⊕compound statements.
Predicate Logic
A predicate is a function that always evaluates to either true or false.
A predicate has the form: Predicate-name( List of Arguments).Ex: x is a positive numberPredicate: positive number (x)Positive number (5)= TruePositive number (-5)= False
Predicate Logic
Ex:
“ 5 is greater than 2”
We define the predicate greater than as:
Greater than( x, y): x>y
P (x, y): x>y
P(2,5)= False
P(5,2)= True
Predicate Logic
1- Uses predicates to represent simple statements.
2- Uses Logical connectives ( V, ^, , , ¬ ) ⊕3- Quantifiers:
Universal quantifier: Existential quantifier: 4- Variables: x, y, z….. .
Predicate Logic
Ex: Consider the statement “ x is greater than 14”.
Predicate: p( x, y): x>y
P( x,14): x>14- There is a value greater than 14 is represented
as .........- All values are greater than 14 is represented as
……….- All values are less than 14 as……………..
Predicate Logic
Ex:
Element x belongs to set A.
B (x, A): Element x belongs to set A.
- Every element in A belongs also to B is represented as: x [ b( x, A) b( x, B)]
Free and Bound variables
The variable x is said to be bound by x or by x if x lies in the scope of the quantifier.
A variable that is not bound by a quantifier is said to be free.
Free and Bound Variables
Ex: Below, describe the scope of each quantifier, and describe which variables are bound and which are free.
x ( p (x) ^ y (t( x, y) ^ r(x)))No free variables.- ¬ x (p(x) ^ y (t(x,y)) V r(z))Z is free.- ¬ x (p(x) ^ y (t(x,y)) V r(y)).Y in t(x,y) is bound but the y in r(y) is free.
Free and Bound Variables
Ex: x [ b( x, A)] b( x, B)
Means: If A is the universe, x belongs to B.
What is the scope of the quantifier? x [ b( x, A)] x [b( x, B)]
It means: If A is the universe then B is the universe.
Predicate Logic
Ex:Assume b(x,y) represents the statement “x belongs to y”. Represent each of the following in predicate logic:- 2 belongs to S.- 1 belongs to A and 2 belongs to B.- All elements in A are positive.- There is an element in A that is not in B.- There is an element in A that is greater then any element in B.- A is a subset in B.
Predicate Logic ( quantifiers)
The statement x s(x) is true iff s is true for every element in the universe.
The statement x s(x) is true iff s is true for at least one element in the universe.
Predicate Logic (quantifiers)
Ex: Suppose Universe: the set of +ve integers s(x) represents “x is an even integer” p(x) represents “x is a prime integer” r(x) represents “ x>2”
Which of the following are true and which are false? x p(x) …. True( try x=2) x p(x) …. False (try x=4) x (p(x) ^ s(x)) … true ( x=2) x (p(x) ^ s(x) ^ r(x)) …false x (s(x) p(x))…false ( try x=4) x (p(x) s(x))…false (try x=3) x (p(x) s(x))… true (x=2) x [(r(x)^s(x))p(x)] …. true(x=2)
Equivalence
Two statements p and q in predicate logic are equivalent if for any universe and for any statements about the universe we substitute for p,q the resulting statements about the universe are equivalent.
Ex:
x(¬ s (x)) is equivalent to ¬ x (s (x))
Equivalence Rules
The following quantified statements are equivalent. x(¬ s (x)) ↔ ¬ x (s (x))- (x s(x)) ^t ↔ x (s(x) ^t)- (x s(x)) ^t ↔ x (s(x) ^t)- (x s(x)) v t ↔ x (s(x) vt)- (x s(x)) v t ↔ x (s(x) v t)- [x p(x)] ^ [x q(x)] ↔ x [p(x) ^ q(x)] - [x p(x)] v [z q(z)] ↔ x [p(x) v q(x)]
Equivalence
Ex:
- [w p(w)] ^ [w q(w)]
- w [p(w) ^ q(w)]
Are they equivalent? Why?
Equivalence
Ex:
- y [x p(x,y)] x [y p(x,y)]
Are they equivalent? Why?
Section 2.2
Proof Methods:- Direct proof- Indirect proofs:
a- Contra positive inference
b- Proof by contradiction
Converses
The statement qp is the converse of the statement pq.
If pq is true, it does not mean that qp is true.
-Ex:-If n is a positive even integer, then n>1. (pq)- 5>1, then 5 is a positive even integer
(qp)....false
Counter-example
To show that a statement is theorem we give a proof.
To show that a statement is false (not theorem), we give a counter-example.
Ex:
“ If n is a positive integer, then n >5”
Counter-example: n=4 (positive and <5)
Direct proof or principle of direct inference (also called modes ponens)
If we know that r is true, and rs is true, we conclude that s is true. A direct proof has the form:
Statement1Statement 2...Statement n
Where statement n is the one we want to prove and each other statement is:
a- a hypothesisb- an accepted mathematical factc- the result of applying direct inference to earlier statements
Direct proof
Ex:
Prove that if the integers n and m are each multiple of 3, then m+n is a multiple of 3.
Note: The word assume precedes the hypothesis and the words ( therefore, then) precedes the inference.
Contra positive Inference
To show that pq, we show that ¬q¬p
Ex:
Prove that for each number n of the universe of positive integers, if n2 >100 then n>10.
Proof by Contradiction
From p and p^¬q¬p we conclude q. If assuming that ¬p leads to contradiction,
the p is true.
Ex:
Prove that if x2 +x-2 =0 then x ≠ 0
Also, see example 15 page 74
Example
Show that if the following statements are true
p
pq
qr
rs
Then s is also true.
(Prove it by both direct and contradiction).