A2 Part A Mathematical Logic Section 1 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 False T F
Composite proposition True or False T Proposition John 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.
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
Truth Tables (I) “P and Q”, denoted by PQ F Sometimes composite proposition is so complicated that its true value is not obvious, we need to study it in a more systematic way I.e. by constructing truth tables.
Truth Table for “P or Q”, denoted by P Q
Truth Table for negation of P, denoted by ~P
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) T F
Proof of ~(P Q) (~P) (~Q) T F
Section 3 Conditional Propositions: If P then Q, denoted by P Q Determine the truth value of the following: If Confucius was Chinese then London is the capital of China. If a man can live without air then the earth will explode at the end of the century. If x = 2 then x2 = 4. If a triangle is isosceles then the base angles are equal. If n2 is an even integer then n is an even integer. Sometimes it is very difficult to determine the truth value of a complicated statement, we therefore need truth table very much.
Truth Table for P Q P Q P Q T F
Definition 3.4 Make the truth tables for these four propositions. Are they equivalent? Definition 3.4 Let P Q be a conditional proposition. This proposition has the following three derivatives(衍生命題): The converse(逆命題) Q P, The inverse(否命題) (~P) (~Q) The contrapositive(逆反命題) (~Q) (~P)
Proof by contrapositive(反證法) P Q (~Q) (~P) Example 1 If 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 that p + 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: If you pass both Physics and Chemistry, then you are able to promote to F.7. If x2 4 and x > 0, then x 2.
Proof by contradiction(歸謬法) (~P) F Class work : Use the method of contradiction to prove that 3 is irrational. 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|q and 2|(p, q) , which is a contradiction.
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 F T T F F F F T T T T T F
~(PQ) ~( (~P)Q) P(~Q) Write down the negation of P Q If today is Sunday, you need not go to school. If I can live without food, then I need not earn money. P (P Q) In the classroom, all students are girls.
Write the negation of: Nobody can answer the question. All triangles having equal bases and equal heights have equal areas. Some people cannot swim. At least there is man who does not like watching television programs. For every positive M, there exists a real number x0 such that x0+logxo>M
Examples of proof by contradiction ~(PQ) ~( (~P)Q) P(~Q) F If x=2, then x is irrational. Proof: Assume that x= 2 and x is not rational, then … 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.6
Illustrative Examples 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=0 i.e.(a+b)2+(b+c)2+(c+d)2+(a-d)2=0 a+b=b+c=c+d=a-d=0 i.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: Nobody can answer the question. For any positive integer n, n + 8 > 0. All students are clever and some of them are lazy. For any even number x, if x is divisible by 3 then x is divisible by 6. 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: If I receive a bonus, I shall have a holiday in Spain. I shall have a holiday in Spain if I receive a bonus. I shall have a holiday in Spain provided that I receive a bonus. I receive a bonus only if I shall have a holiday in Spain. Receiving a bonus is a sufficient condition for a holiday in Spain. Having a holiday in Spain is a necessary condition for receiving a bonus.
Universal Quantifier : for all Existential Quantifier: for some Classwork:1.Translate the propositions on P.64 Q4 to symbols. 2.Negate the above Propositions. Universal Quantifier : for all Existential Quantifier: for some 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) The equation xn+yn=zn has no integral solutions for all integers n 3. (integer n 3)(xn+yn=zn has no integral solutions.) For some real numbers n, if n2=4 then n = 2. (real n)(n2=4 n = 2)
Section 4 Biconditional Propositions Definition 4.1 Let 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)
B Example 4.1 h A m n C 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.
Theorem 4.1 If (PQ)(Q R) then P R. If (PQ)(Q R) then P R. P Q Q P Group discussion: Prove proposition 1-3
If (PQ)(Q R) then P R. Proof: P R T F
Exercise on Logic Prove that if 3|n2 then 3|n. 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. Prove that 3 is irrational. Prove that log2 is irrational. Prove that if 0 x < y for any real number y, then x = 0. 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. The product of any five consecutive natural numbers is not a perfect square.