Solution i If a function is integrable then it is differentiable ii If a

Solution i if a function is integrable then it is

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Solution (i) If a function is integrable, then it is differentiable. (ii) If a function is not integrable, then it is not differentiable. (iii) If the function is not differentiable, then it is not integrable. Definition 1.2.4. (Biconditional Statement): Let P and Q be any two statements. The compound statement “P if and only if Q” is called the biconditional statement of statements P and Q. Symbolically we denote this by P Q . The “if and only if” can be abbreviated as “iff”. We can illustrate the truth table of P Q as follows: P Q P Q 1 1 1 1 0 0 0 1 0 0 0 1 9
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1.3 Laws of Logic Definition 1.3.1. (Logically Equivalent Statements): Two statements, P and Q, are said to be logically equivalent if their truth values coincide . We denote this by P Q . Example 1.3.2. Show the following logically equivalent statements using truth table. (i) ( P Q ) ≡ ¬ P Q (ii) ¬ ( P Q ) ≡ ¬ P ∨ ¬ Q (iii) ¬ ( P Q ) ≡ ¬ P ∧ ¬ Q (iv) ¬ ( P Q ) P ∧ ¬ Q Solutions We will do ( i ) and the rest will be left as an exercise. P Q ¬ P P Q ¬ P Q 1 1 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 1 1 1 As we can see the last two columns have the same truth values so the two statements are logically equivalent. Definition 1.3.3. (Tautology): Tautology is a statement that is always true . Definition 1.3.4. (Contradiction): Contradiction is a statement that is al- ways false . Example 1.3.5. Determine whether the following statements are tautology, contradiction or neither of the two. (i) (( P Q ) ∧ ¬ Q ) ⇒ ¬ P 10
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(ii) ¬ (( P Q ) P ) (iii) P ( ¬ P ( P Q )) Solution (i) (( P Q ) ∧ ¬ Q ) ⇒ ¬ P P Q ¬ P ¬ Q P Q ( P Q ) ∧ ¬ Q (( P Q ) ∧ ¬ Q ) ⇒ ¬ P 1 1 0 0 1 0 1 1 0 0 1 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 Last column shows the statement is always true, hence tautology . (ii) ¬ (( P Q ) P ) P Q P Q ( P Q ) P ¬ (( P Q ) P ) 1 1 1 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 Last column shows the statement is always false, hence contadiction . (iii) P ( ¬ P ( P Q )) P Q ¬ P P Q ¬ P ( P Q ) P ( ¬ P ( P Q )) 1 1 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 0 1 From last column we can conclude the statement is neither tautology nor contradiction . 11
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Logically Equivalences Equivalence Name P 0 0 P 1 1 Domination law P 1 P P 0 P Identity law P P P P P P Idempotent law ¬ ( ¬ P ) P Double negation law P Q Q P P Q Q P Commutative laws ( P Q ) R P ( Q R ) ( P Q ) R P ( Q R ) Association laws P ( Q R ) ( P Q ) ( P R ) P ( Q R ) ( P Q ) ( P R ) Distributive laws ¬ ( P Q ) ≡ ¬ P ∨ ¬ Q ¬ ( P Q ) ≡ ¬ P ∧ ¬ Q De Morgan’s laws P ∧ ¬ P F P ∨ ¬ P T Negation law Tutorials 1.3.6. Determine whether the following statements are tautology, contradiction or neither of the two. (i) ( P Q ) P (ii) ( P ( P Q )) Q (iii) (( ¬ Q ) ( P Q )) ( ¬ P ) (iv) ( P Q ) ( P Q ) (v) ( Q ( P Q )) P (vi) ( P Q ) ( Q P ) 12
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1.4 Arguments Definition 1.4.1. (Valid Argument): An argument is called a valid argu- ment if the conjunction of hypotheses implies the conclusion. Otherwise the argument is said to be invalid or also called fallacy. We can prove the validity of the argument in two ways namely: (a) By following the flow of the argument, (b) by using truth tables.
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