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 twostatements.The compound statement“P if and only if Q”is called thebiconditional statement of statements P and Q. Symbolically we denote this byP⇔Q. The “if and only if” can be abbreviated as “iff”.We can illustrate the truth table ofP⇔Qas follows:PQP⇔Q1111000100019
1.3Laws of LogicDefinition 1.3.1.(Logically Equivalent Statements):Two statements, Pand Q, are said to be logically equivalent if theirtruth values coincide. Wedenote this byP≡Q.Example 1.3.2.Show the following logically equivalent statements using truthtable.(i) (P⇒Q)≡ ¬P∨Q(ii)¬(P∧Q)≡ ¬P∨ ¬Q(iii)¬(P∨Q)≡ ¬P∧ ¬Q(iv)¬(P⇒Q)≡P∧ ¬QSolutionsWe will do (i) and the rest will be left as an exercise.PQ¬PP⇒Q¬P∨Q11011100000111100111As we can see the last two columns have the same truth values so the twostatements are logically equivalent.Definition 1.3.3.(Tautology):Tautology is a statement that isalways true.Definition 1.3.4.(Contradiction):Contradiction is a statement that isal-ways false.Example 1.3.5.Determine whether the following statements are tautology,contradiction or neither of the two.(i) ((P⇒Q)∧ ¬Q)⇒ ¬P10
(ii)¬((P∧Q)⇒P)(iii)P⇔(¬P∧(P∨Q))Solution(i) ((P⇒Q)∧ ¬Q)⇒ ¬PPQ¬P¬QP⇒Q(P⇒Q)∧ ¬Q((P⇒Q)∧ ¬Q)⇒ ¬P1100101100100101101010011111Last column shows the statement is always true, hencetautology.(ii)¬((P∧Q)⇒P)PQP∧Q(P∧Q)⇒P¬((P∧Q)⇒P)11110100100101000010Last column shows the statement is always false, hencecontadiction.(iii)P⇔(¬P∧(P∨Q))PQ¬PP∨Q¬P∧(P∨Q)P⇔(¬P∧(P∨Q))110100100100011110001001From last column we can conclude the statement isneither tautologynor contradiction.11
Logically EquivalencesEquivalenceNameP∧0≡0P∨1≡1Domination lawP∧1≡PP∨0≡PIdentity lawP∧P≡PP∨P≡PIdempotent law¬(¬P)≡PDouble negation lawP∧Q≡Q∧PP∨Q≡Q∨PCommutative laws(P∧Q)∧R≡P∧(Q∧R)(P∨Q)∨R≡P∨(Q∨R)Association lawsP∧(Q∨R)≡(P∧Q)∨(P∧R)P∨(Q∧R)≡(P∨Q)∧(P∨R)Distributive laws¬(P∧Q)≡ ¬P∨ ¬Q¬(P∨Q)≡ ¬P∧ ¬QDe Morgan’s lawsP∧ ¬P≡FP∨ ¬P≡TNegation lawTutorials 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
1.4ArgumentsDefinition 1.4.1.(Valid Argument):An argument is called avalid argu-mentif the conjunction of hypotheses implies the conclusion.Otherwise theargument is said to beinvalidor also calledfallacy.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.