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Mathematical Reasoning S3S

hat is logic ? In general , the goal of study of Logic is to construct good or sound arguments , and to recognise bad or unsound arguments . Thus , Logic is the science of reasoning .

The first great treatises on logic were written by the Greek philosopher Aristotle. They were a collection of rules for deductive reasoning that were intended to serve as a basis for the study

of every branch of knowledge. In the 17 century , the German philosopher and mathematician Gottfried Leibniz conceived the idea of using symbols to mechanize the process of deductive reasoning in much the same way that algebraic notation had mechanized the process of reasoning about numbers and their relationships. The first one to employ mathematical methods in the study of Logic was English mathematician George Boole (1815-1864) , who founded the modern subject of symbolic logic . With research continuing to the present day , symbolic logic has provided , among other things , the theoretical basis for many areas of computer science such as digital logic circuit design , relational database theory , and artificial intelligent.

1.1 Propositions (or Statements1)

• A proposition is a sentence that can be assigned a truth value : true or false , but not both .

Example : (1) 1 + 1 = 2 (2) 12 may be written as the sum of two prime numbers . (3) All prime numbers are odd numbers . (4) The sum of x and y is greater than 0 . (5) For any natural numbers x and y , the sum of x and y is greater than 0. Of these the first three are propositions . (1) , (2) , (5) are true and (3) is false . There is no ambiguity regarding these sentences . Therefore they are statements . (4) is not a statement since we cannot determine whether it is true or false , unless we know what x and y are . For example , it is false where x =1 , y = -3 and true when x =1 and y = 0 . While dealing with statements , we usually denote them by small letters p,q,r,…. For examples , we denote the statements “ 8 is less than 6 ” by p. This is written as p : 8 is less than 6 .

Example 1 : Check whether the following sentences are statements . (i) Open the door . (vi) All roses are white .

(ii) π is a rational number . (vii) How tall is Alvin ?

(iii) x2 + 5x + 6 = (x+2) (x-3) (viii) He is a mathematician .

(iv) A triangle has four sides . (ix ) x + 7 = 23

(v) Girls are intelligent than boys . (x ) The base angles of ∆ABC are equal.

Truth value The truth or falsity of a statement is called its truth value .

The value of a truth proposition is 1 The value of a false proposition is 0.

1 Commands , questions and exclamations are not statements .A sentence which is both true and false simultaneously is not a statement , rather , it is a paradox .

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Example 2: Find the value of p: ∀ a ∈ ℝ , if a3 > 0 , then a > 0 . q: the line 3x – 4y = 1 passing through the origin . r: ∀ x ∈ ℝ , 2x + 1 > x .

1.2 Negation of a statement Let p be any statement. Then the statement expressing denial of p is called negation of p . Negation of p is formed by writing it is false…before p . Negation is formed by inserting the word ‘ not ’ p . Negation of statement p is denoted as ~ p . Remark : the negative of the statements that contain the words like “for all” , “there exists”, “some” or “for every” can be tricky . Example : p: All mathematicians are men . q: There exists a dog which does not bite . ~ p : Not all mathematicians are men . ~q: There does not exist a dog which does not bite.

The truth values for negation are summarized in a truth table .

Example 3 : Write the negation of the following statements : (a) p : Every natural number is an integer . (b) q : All triangles are not equilateral triangles. (c) r : For every positive real number x the number x-1 is also positive . (d) s : There exists a number x such that 0 < x < 1 .

1.2.3 CONNECTING WORDS Simple proposition is a statement which cannot be broken into two or more statements . For example :

(i) Every set is a finite set (ii) Roses are red are all simple statements . If two or more simple statements are combined by the use of connecting words as : “and” , “or” , “if … then” , “if and only if ” , then the resulting statement is called a compound proposition . The truth values of compound statements would depend upon the truth values of the constituent statements .

Words that are used to modify a statement or to combine two statements are called logical connective .

p ~p

1 0

0 1

Connecting Words Symbol Compound statement

AND ∧ Conjunction

OR ∨ Disjunction

IF ….. THEN ⟶ Conditional statement

IF AND ONLY IF ⟷ Biconditional statement

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(A) Conjunction If two statements are combined by using the connecting word ‘ and ’ then the resulting statement is called conjunction . The conjunctions of statements p and q is denoted by p ∧ q . In truth table

For example , let p : It is cold . q : It is raining . then p ∧ q : It is cold and raining .

Example : Let p: 2 divides 4

q : 8 + 2 = 12

then p ∧ q :

∴ p ∧ q =

(B) Disjunction If two statements are combined by using the connecting word ‘ or ’ , then the resulting statement is called a disjunction . The disjunction of two statements p and q is denoted by p ∨ q .

In truth table : For example , let p : 8 ≤ 10 q : 4 is an integer . then p ∨ q : 8 ≤ 10 or 4 is an integer

Example : Determine the truth value of the compound statement .

(a) 2 is natural number or 2 is even number . (b) 2 is a rational number or an irrational number.

(C) Implications “If then , ” “only if ” and “if and only if ” are known as implications . If p and q are two statements forming the implication “ if p then q ” , then we denote this implication by “ p → q ” . The symbol → stands for implies .

For example , let p : 2 + 5 = 7 and q : 9 is an integer , then their conditional statement p → q denotes the statement : “ If 2 + 5 = 7 , then 9 is an integer. ”

If then

The conditional statement p→ q can be read in different ways as : (a) p implies q (b) p is sufficient for q (c) q is necessary for p (d) p only if q

Example 4 : Write the following statements implication statement and hence determine its truth value .

(a) p : a2 + b2 ≥ 2ab (b) p: 3 divides 15 (c) p: For every real x , |x| < x is hold.

q : a > 0 , b > 0 q : 5-1 = 4 q: For every real x , |x| > 0 is hold .

p q p ∧ q

1 1 1

1 0 0

0 1 0

0 0 0

p q p ∨ q

1 1 1

1 0 1

0 1 1

0 0 0

p q p → q

1 1 1

1 0 0

0 1 1

0 0 1

Antecedent Premises Hypothesis

Consequent Conclusion

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Example 5 : Let p : x = 3 and q: x2 = 9 respectively , describe the following conditional statements :

(i) p → q (ii) q→p (converse proposition) (iii) ~p→~q (inverse proposition)

(iv) ~q → ~p (contrapositive proposition)

Example 6: Verify by method of contraction p : 7 is irrational .

(D) Equivalence The equivalence of two statements p and q is “ p if and only if q ” and is written as p ↔ q . p ↔ q is true only when either both p and q are true or false . p ↔ q is also known as biconditional or double implication statement .

• p is equivalent to q • p implies and is implied by q • p holds if q and conversely • p is a necessary and sufficient condition for q

The phrase “ if and only if ” appears so often in Mathematics that it is customary to abbreviate it as “ iff ” .

2. Evaluating the Truth of More General Compound Proposition

p q p ⟷ q

1 1 1

1 0 0

0 1 0

0 0 1

Definition: A statement form (or proposition form) is an expression made up of statement variables (such as p,q and r ) and logical connectives (such as ~ , ∧ and ∨ ) that becomes a statement when actual statements are substituted for the component statement variables.

The truth table for a given statement form displays the truth values that correspond to all possible combinations of truth values for its component statement variables.

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Remarks : To compute the truth values for a statement form , first evaluate the expression within the innermost parentheses , then evaluate the expressions within the next innermost set of parentheses , and so forth until you have the truth values for the complete expression .

Example 7 : Construct a truth table of the following compound statements .

(a) ( ) ( )p q p∧ ∧ (b) ( ~ ) ( )p q p q∨ → ∧

(c) (~ )p q p∨ ∧ (d) ( ) ~p q r∧ ∨

Logically Equivalences Compound propositions that have the same truth values in all possible cases are called logically equivalent. The logical equivalence of statement forms P and Q is denoted by writing P ≡ Q.

Definition :

• Tautology : A compound proposition that is always true , no matter what the truth values of the propositional variables that occur in it , is called a tautology . Contradiction : A compound proposition that is always false is called contradiction.

Examples of a Tautology and a Contradiction

p ~p p ∨ ~p (Tautology) p ∧ ~p (contradiction)

1 0 1 0

0 1 1 0

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Example 8: Use the truth table to determine which is tautology or contradiction .

1) (~ ) ( ~ )p q p q∨ ∨ ∧ 【ans : tautology】

2) ~ [( ) ( )]p q p q∧ → ∨ 【ans : contradiction】

\ Example 9: If ( )p q r→ ∨ is false , find the truth values of p , q , r .

Example 10: Let p , q , r are three propositions . Use truth table to prove that ~ ( ( ~ )) (~ )p q r p q r→ ∧ ≡ ∧ ∨

Example 11: Show that the conditional statements [~ ( )]p p q q∧ ∨ → is tautology by using truth tables.

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Argument An argument in propositional logic is a sequence of propositions . All but the final proposition in the argument are called premises and the final proposition is called the conclusion . From the propositions P1 , P2 , …. , Pn , leads to the conclusion Q , this can be summarised as P1 , P2 , P3 , … , Pn ; ∴Q

If the propositions P1 , P2 , P3,…, Pn are all true , and the conclusion are also true , then the argument is valid .

From the definition of a valid argument form we see that the argument form with premises

P1 , P2 , … , Pn and conclusion Q is valid , when (P1 ∧ P2 ∧ … ∧ Pn ) → Q is a tautology .

Step for testing an Argument Form for Validity • Identify the premises P1 , P2 , … , Pn and conclusion Q of the argument form. • Construct a truth table showing the truth values of all the premises and the conclusion . • If (P1 ∧ P2

∧ … ∧ Pn ) → Q is a tautology then the argument P1 , P2 , P3 , … , Pn ; ∴Q is valid otherwise it is invalid .

Example 12: Test the validity of ~ , ; ~qp q p→ ∴

Example 13: Test the validity of : Only if the quadrilateral is a parallelogram the opposite are equal . The opposite sides are not equal . Therefore , the quadrilateral is not a parallelogram . 【ans: invalid】

Example 14: Test the validity of the following argument form : , ~ , ;p q p q p r r∨ → → ∴ 【invalid】

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Review Question : 1. If p ↔ q ≡ ( p→q ) ^ ( q→p) , find the truth table of p↔q . 2. If p ⨁ q is defined as (p ⋁ q )⋀∼(p⋀q) , Determine whether (p ⊕ q ) ⋀ r ≡ ( p ⋀ r ) ⨁ ( q ⋀ r ) . 3. Show that [( p → q ) ⋀ ( q→ r ) ] → ( p → r ) is tautology . 4. Determine the validity of the following argument : If 7 is less than 4 , then 7 is not a prime number . 7 is not less than 4 . 7 is a prime number .

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5. Test the validity of the following argument : p → ~q , r→ p , q ; ∴ ~ r . 6. Given that p , q and r are three propositions . By using a truth table , prove that ( p → r ) ⋁ ( q→ r ) ≡ (( p⋀ q ) → r ) . [2009 AM Que 4(b) ]

7. Let p,q and r be three propositions , and the compound proposition p q↓ is defined as ~ ( )p q p q↓ = ∨ .

(i) Construct a truth table of p q↓

(ii) Use the truth table in (i) , construct a truth table to determine whether the propositions ( )p q r↓ ↓ and the proposition ( )r p q→ ∨ are equivalent .