A2 Part A Mathematical Logic

Simple Proposition (Statements)Hong Kong is an international city.Opposite angles of a parallelogram are equal.Blind men can see.2+3=7.

True or FalseT

T

FF

Section 1

Composite proposition

PropositionJohn is a boy and Mary is a girl.Snow is white or the sun rises from the West.If today is Friday then the earth is spherical.

True or FalseT

T

T

Example 1.1 Determine the truth values of the following:

Paris is the capital of France and 2+2=5.Christine is a girl and the sun rises from the East.Yellow river is in Europe or snow is black.22

F

T

FT

Truth Tables(I) “P and Q”, denoted by PQ

P Q PQ

T T T

T F F

F T F

F F F

Truth Table for “P or Q”,denoted by P Q

P Q P Q

T T T

T F T

F T T

F F F

Truth Table for negation of P,denoted by ~P

P ~P

T F

F T

Negate the following statements:

(i) The sun is spherical and the plane

can fly.

(ii) London is not the capital of China

or the house is made of wood.

Section 2 Equivalence of Two Propositions

Two propositions with the same components P, Q, R,… are said to be logically equivalent(or equivalent) if they have the same truth value for any truth values of their components.

De Morgan’s Law

Let P, Q be two propositions, then

(I) ~(PQ)

(II) ~(PQ)

Proof of ~(PQ) (~P)(~Q)

P Q PQ ~(PQ) ~P ~Q (~P)(~Q)

T T

T F

F T

F F

Proof of ~(P Q) (~P) (~Q)

P Q P Q ~(P Q) ~P ~Q (~P) (~Q)

T T T F F F F

T F T F F T F

F T T F T F F

F F F T T T T

Section 3 Conditional Propositions:If P then Q, denoted by P Q

Determine the truth value of the following:1. If Confucius was Chinese then London is the

capital of China.2. If a man can live without air then the earth

will explode at the end of the century.3. If x = 2 then x2 = 4.4. If a triangle is isosceles then the base angles

are equal.5. If n2 is an even integer then n is an even

integer.

Truth Table for P Q

P Q P Q

T T T

T F F

F T T

F F T

Definition 3.4

Let P Q be a conditional proposition. This proposition has the following three derivatives( 衍生命題 ):

1. The converse( 逆命題 ) Q P,2. The inverse( 否命題 ) (~P) (~Q)3. The contrapositive( 逆反命題 ) (~Q)

(~P)

1. Make the truth tables for these four propositions.

2. Are they equivalent?

1. Make the truth tables for these four propositions.

2. Are they equivalent?

Proof by contrapositive( 反證法 )

P Q (~Q) (~P)Example 1If n2 is an even integer then n is an even integer.Proof:If n is odd, then n = 2k + 1 and

(2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1 is odd. Thus by contrapositive, the proposition is correct.

Example 2

Given that p and m are real numbers such that p3+m3=2, prove thatp + m 2.Proof:Assume that p+m>2, then p3+m3>(2-m)3+m3=6m2-12m+8=6(m-1)2+2>2. Thus, by contrapositive, p + m 2.

Write down the contrapositive of the following propositions:

1. If you pass both Physics and Chemistry, then you are able to promote to F.7.

2. If x2 4 and x > 0, then x 2.

Proof by contradiction( 歸謬法 )(~P) F

Example 3.3 Use the method of contradiction to prove that 2

is irrational.Proof: Suppose that 2 is not irrational, then 2 =

p/q for some natural numbers p, q where (p, q) = 1. Since

2 =p2/q2, therefore 2q2=p2. This implies that 2|p2 and hence 2|p. So p=2k for some integer k. Putting it back to 2q2=p2,

2q2=(2k)2 i.e. q2=2k2. Again, we have 2|qand 2|(p, q) , which is a contradiction.

Class work : Use the method of contradiction to prove that 3 is irrational.Class work : Use the method of contradiction to prove that 3 is irrational.

Theorem (proved by Euclid): There are infinitely many prime numbers.

Proof:Assume there are only n prime numbers, say p1, p2, p3,…,pn. Now construct a new number p= p1p2p3…pn + 1, then p is a new prime number since p is not divisible by pi’s and p > pi’s. This leads to a contradiction that p1, p2, p3,…,pn are the only prime numbers. So there are infinitely many prime numbers.

Sometimes the proposition is conditional i.e. PQ,

We need to negate it in order to prove it by contradiction.i.e. ~ (PQ) F.

But ~ (PQ) ?

(Hint: Find an equivalent statement for PQ which involves P, Q, ~ and .)

PQ (~P)Q

P Q P Q (~P ) Q

T T T F T T

T F F F F F

F T T T T T

F F T T T F

Write down the negation of 1. P Q2. If today is Sunday, you need not go to

school.3. If I can live without food, then I need

not earn money.4. P (P Q)5. In the classroom, all students are girls.

~(PQ) ~( (~P)Q) P(~Q)

Write the negation of:

6. Nobody can answer the question.7. All triangles having equal bases and

equal heights have equal areas.8. Some people cannot swim.9. At least there is man who does not

like watching television programs.10.For every positive M, there exists a

real number x0 such that x0+logxo>M

Examples of proof by contradictionExamples of proof by contradiction

~(PQ) ~( (~P)Q) P(~Q) F

1. If x=2, then x is irrational.Proof: Assume that x= 2 and x is not rational, then …

2 If x=n and y=n+1, then x and y are relatively prime.Proof: Assume that x=n and y=n+1 and x and y have common factor other than 1, say f, then n=fg and n+1=fh. So 1 = f(h-g) and hence f=1, which is a contradiction. Thus the proposition is true.

P.65, Q.6P.65, Q.6

Illustrative Examples

3. If ABC is a acute triangle and A>B>C, prove that B> 45.Proof:Assume that ABC is a acute triangle and B 45, then C < 45.But A=180 - B- C > 90 leads to a contradiction that ABC is a acute triangle. Thus,by the method of contradiction, B> 45.

4. Given that a, b,c and d are real numbers and ad-bc=1, prove that a2+b2+c2+d2+ab+cd1.

Proof:Assume that a, b,c and d are real numbers and ad-bc=1, but a2+b2+c2+d2+ab+cd=1, then a2+b2+c2+d2+ab+cd=ad-bc. Multiplying it by 2, we get 2a2+2b2+2c2+2d2+2ab+2cd-2ad+2bc=0i.e.(a+b)2+(b+c)2+(c+d)2+(a-d)2=0a+b=b+c=c+d=a-d=0i.e.a=b=c=d=0, which contradict to that ad-cd=1. Thus, by the method of contradiction, a2+b2+c2+d2+ab+cd 1.

Write the negation of:

7. Nobody can answer the question.8. For any positive integer n, n + 8 > 0. 9. All students are clever and some of

them are lazy.10.For any even number x, if x is

divisible by 3 then x is divisible by 6.11.There exist natural numbers p and q

such that 2 = p/q.

Definition 3.2

When the conditional proposition P Q is always true, we write P Q and read as P implies Q.For instance, it is correct to write

“x = 2 x2 = 4”, but incorrect to write “x + a = b x = a + b”

Definition 3.3

Let P Q be a conditional proposition. Then P is called the sufficient condition ( 充分條件 ) for Q,

and Q is the necessary condition( 必要條件 ) for P.

Pick out the different one from the following statements:1. If I receive a bonus, I shall have a holiday in

Spain.2. I shall have a holiday in Spain if I receive a

bonus.3. I shall have a holiday in Spain provided that I

receive a bonus.4. I receive a bonus only if I shall have a holiday in

Spain.5. Receiving a bonus is a sufficient condition for a

holiday in Spain.6. Having a holiday in Spain is a necessary

condition for receiving a bonus.

Universal Quantifier : for allExistential Quantifier: for some1. Some birds are white.

In symbol, (bird B)(B is white)2. For any integer n , the equation x2-nx+1=0

must have a real solution.(integer n)(x2-nx+1=0 has a real

solution)3. The equation xn+yn=zn has no integral

solutions for all integers n 3.(integer n 3)(xn+yn=zn has no integral

solutions.)4. For some real numbers n, if n2=4 then n = 2.

(real n)(n2=4 n = 2)

Classwork:1.Translate the propositions on P.64 Q4 to symbols. 2.Negate the above Propositions.

Classwork:1.Translate the propositions on P.64 Q4 to symbols. 2.Negate the above Propositions.

Section 4 Biconditional Propositions

Definition 4.1Let P and Q be two propositions.The biconditional proposition PQ (read as “P if and only if Q”) is defined as P Q (PQ) (QP)

Complete the Truth Table of P Q (PQ) (QP)

P Q PQ QP (PQ) (QP)

T T

T F

F T

F F

Example 4.1

In the Figure, P is a point on AC such that BPAC, PA = m, PB = h and PC = n. Prove that h2= mn iff ABC = 90.

h

m nA

B

C

Theorem 4.1

1. If (PQ)(Q R) then P R.

2. If (PQ)(Q R) then P R.

3. P Q Q PGroup discussion: Prove proposition 1-3

If (PQ)(Q R) then P R.Proof:

P Q R PQ Q R P R

T T T T T T T T

F T T T T T T T

F F T T T T T T

F F F T T T T T

Exercise on Logic1. Prove that if 3|n2 then 3|n.2. Prove that for any real numbers a, b, c and d, if a +

bi = c + di then a = c and b=d, where i2= -1.3. Prove that 3 is irrational.4. Prove that log2 is irrational.5. Prove that if 0 x < y for any real number y, then x

= 0.6. Prove that if f(x) is not identically zero and f(xy) =

f(x)f(y), the f(x) 0 for any non-zero real number x. 7. The product of any five consecutive natural

numbers is not a perfect square.