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Click Here To Download Review Exercise Ex : 6.2 Ex : 6.3 Ex : 6.4 Ex : 6.5 Ex: 6.6 MCQs and Shorts

Click Here To Download Review Exercise Ex : 6.2 Ex : 6.3 Ex : 6.4 Ex : 6.5 Ex: 6.6 MCQs and Shorts

Click Here To Download Review Exercise Ex : 6.2 Ex : 6.3 Ex : 6.4 Ex : 6.5 Ex: 6.6 MCQs and Shorts

Click Here To Download Review Exercise Ex : 6.2 Ex : 6.3 Ex : 6.4 Ex : 6.5 Ex: 6.6 MCQs and Shorts

Click Here To Download Review Exercise Ex : 6.2 Ex : 6.3 Ex : 6.4 Ex : 6.5 Ex: 6.6 MCQs and Shorts

Click Here To Download Review Exercise Ex : 6.2 Ex : 6.3 Ex : 6.4 Ex : 6.5 Ex: 6.6 MCQs and Shorts

Click Here To Download Review Exercise Ex : 6.2 Ex : 6.3 Ex : 6.4 Ex : 6.5 Ex: 6.6 MCQs and Shorts

Click Here To Download Review Exercise Ex : 6.2 Ex : 6.3 Ex : 6.4 Ex : 6.5 Ex: 6.6 MCQs and Shorts 1. Father of formal logic – AristotleAristotle ne sabse pehle logic ko systematic tareeqe se organize kiya, isliye usay “father of formal logic” kaha jata hai. 2. Boolean algebra – George BooleGeorge Boole ne mathematics ko logic ke saath jorha aur ek naya system banaya jisay Boolean algebra kehte hain (True/False ya 1/0 par based). 3. Statement kya hoti haiWoh mathematical expression jo ya to true ho ya false ho, lekin dono nahi — usay Statement (ya Proposition) kehte hain. 4. Statement ka doosra naamStatement ko Proposition bhi kaha jata hai — dono terms same cheez ke liye use hoti hain. 5. “Open the door!” statement kyun nahiYeh ek command/order hai, iska koi truth value (true/false) nahi ho sakta, isliye yeh statement nahi hai. 6. Truth value ka doosra naamKisi statement ke true ya false hone ko “Truth value” ya “Logical value” dono kaha ja sakta hai — dono correct hain. 7. “And” ka symbol – ∧Logic mein “and” (conjunction) ko ∧ symbol se represent karte hain. 8. “Or” ka symbol – ∨Logic mein “or” (disjunction) ko ∨ symbol se represent karte hain. 9. Negation ka symbol – ~“Not” ko negation kehte hain aur iska symbol ~ (tilde) hota hai. 10. → ka matlabYeh symbol “if…then” (conditional/implication) ko represent karta hai, jaise p → q ka matlab “agar p to q”. 11. ↔ ka matlabYeh symbol Biconditional ko represent karta hai, jiska matlab hai “p if and only if q”. 12. p ki negation likhne ka tareeqap ki negation ko ~p ya p’ dono tareeqon se likha ja sakta hai. 13. Conjunction ka symbolp aur q ki conjunction ko p ∧ q likhte hain. 14. p ∧ q kab true hoti haiConjunction (AND) sirf tab true hoti hai jab dono statements (p aur q) true hon. Agar ek bhi false ho to poori statement false ho jati hai. 15. p true, q false ho to p ∧ qChunke AND ke liye dono ka true hona zaroori hai, aur yahan q false hai, isliye p ∧ q False hogi. 16. Disjunction true kab hoti haip ∨ q (OR) us waqt true hoti hai jab kam az kam ek statement true ho. 17. p ∨ q false kab hoti haiOR sirf us waqt false hoti hai jab dono statements false hon. 18. p false, q true ho to p ∨ qChunke q true hai, aur OR ke liye sirf ek ka true hona kaafi hai, isliye p ∨ q True hogi. 19. Conditional false kab hota haip → q sirf us waqt false hota hai jab antecedent (p) true ho aur consequent (q) false ho. 20. p → q mein p ka naamp ko Antecedent ya Hypothesis kehte hain — yeh “agar” wala part hota hai. 21. p → q mein q ka naamq ko Consequent ya Conclusion kehte hain — yeh “to” wala part hota hai. 22. p → q kab false hota hai (except case)Conditional sirf tab false hota hai jab p True ho aur q False ho — baqi tamam combinations mein yeh true hota hai. 23. Biconditional kab true hoti haip ↔ q tab true hoti hai jab p aur q dono ka truth value same ho (dono true ya dono false). 24. p ↔ q ko kaise parhte hainIsay “p if and only if q” parha jata hai. 25. ~p → ~q kis ka Inverse haiOriginal statement p → q ka Inverse banane ke liye dono parts ki negation karte hain: ~p → ~q. 26. q → p kya kehlata haiOriginal p → q ke antecedent aur consequent ko switch karne se Converse banta hai: q → p. 27. ~q → ~p kis ka Contrapositive haiAntecedent aur consequent ko switch karke phir negate karne se Contrapositive banta hai: ~q → ~p. 28. Contrapositive hamesha kya hota haiContrapositive apni original statement ke logically equivalent hota hai — matlab dono ka truth value hamesha same rehta hai. 29. Truth table kya hoti haiWoh table jo compound statement ke tamam possible truth value combinations dikhati hai, usay Truth Table kehte hain. 30. n variables ke liye rows ki tadadFormula hai 2ⁿ — jitne variables hon, 2 ko unki power tak raise karte hain. 31. 2 variables (p,q) ke liye rows2² = 4 rows honge. 32. Hamesha true wali statementAisi statement jo har condition mein true rahe, usay Tautology kehte hain. 33. Hamesha false wali statementAisi statement jo har condition mein false rahe, usay Contradiction kehte hain. 34. Kabhi true kabhi false wali statementAisi statement jiska truth value components par depend karta hai (sometimes true, sometimes false), usay Contingency kehte hain. 35. p ∨ ~pYeh hamesha true hoti hai kyunke p ya to true hoga ya false, aur dono cases mein OR true ban jata hai. Isliye yeh Tautology hai. 36. p ∧ ~pYeh hamesha false hoti hai kyunke p aur uski negation kabhi bhi ek saath true nahi ho sakte. Isliye yeh Contradiction hai. 37. Same truth value wali statementsDo statements jinki truth table ki har row mein same value ho, unhe Logically Equivalent Statements kehte hain. 38. De Morgan’s Law – ~(p ∧ q)~(p ∧ q) barabar hota hai ~p ∨ ~q ke — “not-and” ko “not-or” mein convert karte hain. 39. De Morgan’s Law – ~(p ∨ q)~(p ∨ q) barabar hota hai ~p ∧ ~q ke — “not-or” ko “not-and” mein convert karte hain. 40. ~(~p) kya hota haiDouble negation apni original value pe wapis aa jati hai, isliye ~(~p) = p. 41. (p→q) ∧ (q→p) ka short formYeh dono conditionals mila kar Biconditional banta hai: p ↔ q. 42. ∀ symbol ka naamIsay Universal Quantifier kehte hain. 43. ∃ symbol ka naamIsay Existential Quantifier kehte hain. 44. ∀x kaise parhte hain“For all x” ya “for every x” parha jata hai. 45. ∃x kaise parhte hain“There exists at least one x” parha jata hai. 46. Argument valid kab hota haiJab conclusion premises se

Click Here To Download Review Exercise Ex : 6.2 Ex : 6.3 Ex : 6.4 Ex : 6.5 Ex: 6.6 MCQs and Shorts Q.1: Multiple Choice Questions (i) Inductive reasoning se related expression → (A) based on repeated experimentsInductive reasoning mein hum baar baar experiments/observations kar ke general conclusion nikalte hain — jaise agar koi cheez baar baar ho rahi ho to hum ussay general rule bana lete hain. (ii) Deductive reasoning ko describe karne wala jumla → (D) draw conclusion from well-known factsDeductive reasoning mein hum pehle se maloom (well-known) facts se ek naya, guaranteed-true conclusion nikalte hain — general se specific ki taraf jate hain. (iii) Konsi statement sahi hai → (C) 22/7 ∉ Q’22/7 ek rational number hai (Q ka member), aur Q’ irrational numbers ka set hai. Isliye 22/7 Q’ mein nahi hai — ye sahi statement hai. Baaki options ghalat hain (integers infinite hote hain, quadrilateral ke angles 360° hote hain 180° nahi, aur isosceles triangles equilateral nahi hote). (iv) “The stove is burning” ki negation → (A) the stove is not burningNegation ka matlab hai statement ko seedha ulta karna — “burning” ki negation “not burning” hai. (v) p aur q ki conjunction (AND) sach kab hoti hai → (B) both p and q are trueAND (∧) sirf tab true hoti hai jab dono statements true hon. (vi) Conditional (if-then) kab false hoti hai → (A) antecedent is true and consequent is false“If p then q” sirf tab false hota hai jab p (antecedent/pehla hissa) true ho lekin q (consequent/doosra hissa) false ho. (vii) q → p ka Contrapositive → (B) ~q → p Wait, let me check — actual formula: Contrapositive of (q→p) is (~p→~q). Answer key shows option (C) ~p → ~q as correct based on standard logic rule. Contrapositive of q → p → (C) ~p → ~qContrapositive banane ka rule hai: order ulta karo aur dono taraf negation lagao. Isliye q→p ka contrapositive ~p→~q banta hai. (viii) “Every integer greater than 2 is a sum of two prime numbers” → (B) conjectureYe statement (Goldbach Conjecture) abhi tak proved nahi hui, sirf believed hai ke sahi hai — isliye ye conjecture hai, theorem nahi. (ix) “A straight line can be drawn between any two points” → (C) axiomYe ek basic accepted fact hai jisay bina proof ke maana jata hai — isliye axiom hai (Euclid’s Axiom). (x) “Sum of interior angles of a triangle is 180°” → (B) theoremYe ek proven statement hai (geometry mein prove hoti hai), isliye theorem hai. Q.2: Converse, Inverse, aur Contrapositive nikalo Basic Rules yaad rakho: Converse: Order ulta karo (q → p) Inverse: Dono taraf negation lagao, order same rakho (~p → ~q) Contrapositive: Order ulta karo AUR negation lagao (~q → ~p) (i) ~p → q: Converse = q → ~p, Inverse = p → ~q, Contrapositive = ~q → p(ii) q → p: Converse = p → q, Inverse = ~q → ~p, Contrapositive = ~p → ~q(iii) ~p → ~q: Converse = ~q → ~p, Inverse = p → q, Contrapositive = q → p(iv) ~q → ~p: Converse = ~p → ~q, Inverse = q → p, Contrapositive = p → q Trick: Hamesha original statement ko table mein rakh kar teeno formulas ek ek kar ke apply karo — mistake kam hoti hai. Q.3: Truth Tables (i) ~(p∨q)∨(~q):Pehle p∨q (OR) nikalo, phir uski negation lo, phir ~q ke sath OR lagao. Har row mein T/F values calculate kar ke final column tak pohanchte hain. (ii) ~(~q∨~p):Pehle ~q aur ~p nikalo, phir unka OR lo, phir poori cheez ki negation lo. (iii) (p∨q)↔(p∧q):Ye biconditional hai — sach tab hogi jab dono taraf (p∨q aur p∧q) ki value same ho, chahe dono true hon ya dono false. Truth table banane ka tareeqa: Hamesha p aur q ke saare 4 combinations (TT, TF, FT, FF) likho, phir step by step column banate jao jab tak final expression tak na pohanch jao. Q.4: Mathematical Statement aur Proof mein farq Mathematical Statement: Koi jumla ya expression jo ya to true ho sakta hai ya false, lekin dono nahi — isay statement kehte hain. Misal: “a = b” ya to sach hai ya jhoot. Mathematical Proof: Ye evidence deti hai ke statement sahi hai, logical steps ke through. Jaise agar koi bacha (Fayyaz) kahe ke “main school gaya tha” to uska father proof mangega (attendance register) — bilkul isi tarah math mein bhi statement ko sach sabit karne ke liye proof chahiye hoti hai. Doosri misal: mobile phone ki warranty claim karne ke liye warranty card proof ki tarah chahiye hota hai. Q.5: Axiom aur Theorem mein farq Axiom: Wo mathematical statement jisay bina kisi proof ke sach maan liya jata hai — ye basic facts hain jo aage ki ideas ki bunyad banate hain. Misalein: “Ek point se infinite lines guzar sakti hain”, “Do points ke beech seedhi line draw ki ja sakti hai” (Euclid Axiom), “Har natural number ka successor hota hai” (Peano Axiom). Theorem: Wo mathematical statement jo already known facts ki madad se prove ki gayi ho. Misalein: “Quadrilateral ke interior angles ka sum 360° hota hai”, Fundamental Theorem of Arithmetic (har integer >1 ko prime numbers ke product ki tarah likha ja sakta hai), Fermat’s Last Theorem. Q.6: Mathematical proofs mein logical reasoning ki importance Logic ek systematic tareeqa hai jis se hum statements ke meanings samajh sakte hain, unki sachai check kar sakte hain, aur known facts se nayi information nikaal sakte hain. Ye problem-solving aur decision-making mein zaroori hai. Misal: Agar kisi ko penicillin ka injection lagne ke baad reaction ho, to wo general conclusion nikal leta hai ke usay penicillin se allergy hai — ye logic ki misal hai jo hum daily life mein use karte hain. Q.7: Axiom, Conjecture, ya Theorem identify karo (i) “Do points ke beech exactly ek seedhi line hoti hai” → AxiomYe geometry ki bunyadi assumption hai jisay bina proof ke sach maana jata hai. (ii) “Har even number >2 do prime

Click Here To Download Review Exercise Ex : 6.2 Ex : 6.3 Ex : 6.4 Ex : 6.5 Ex: 6.6 MCQs and Shorts