1. Father of formal logic – Aristotle
Aristotle ne sabse pehle logic ko systematic tareeqe se organize kiya, isliye usay “father of formal logic” kaha jata hai.
2. Boolean algebra – George Boole
George 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 hai
Woh mathematical expression jo ya to true ho ya false ho, lekin dono nahi — usay Statement (ya Proposition) kehte hain.
4. Statement ka doosra naam
Statement ko Proposition bhi kaha jata hai — dono terms same cheez ke liye use hoti hain.
5. “Open the door!” statement kyun nahi
Yeh ek command/order hai, iska koi truth value (true/false) nahi ho sakta, isliye yeh statement nahi hai.
6. Truth value ka doosra naam
Kisi 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 matlab
Yeh symbol “if…then” (conditional/implication) ko represent karta hai, jaise p → q ka matlab “agar p to q”.
11. ↔ ka matlab
Yeh symbol Biconditional ko represent karta hai, jiska matlab hai “p if and only if q”.
12. p ki negation likhne ka tareeqa
p ki negation ko ~p ya p’ dono tareeqon se likha ja sakta hai.
13. Conjunction ka symbol
p aur q ki conjunction ko p ∧ q likhte hain.
14. p ∧ q kab true hoti hai
Conjunction (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 ∧ q
Chunke AND ke liye dono ka true hona zaroori hai, aur yahan q false hai, isliye p ∧ q False hogi.
16. Disjunction true kab hoti hai
p ∨ q (OR) us waqt true hoti hai jab kam az kam ek statement true ho.
17. p ∨ q false kab hoti hai
OR sirf us waqt false hoti hai jab dono statements false hon.
18. p false, q true ho to p ∨ q
Chunke q true hai, aur OR ke liye sirf ek ka true hona kaafi hai, isliye p ∨ q True hogi.
19. Conditional false kab hota hai
p → q sirf us waqt false hota hai jab antecedent (p) true ho aur consequent (q) false ho.
20. p → q mein p ka naam
p ko Antecedent ya Hypothesis kehte hain — yeh “agar” wala part hota hai.
21. p → q mein q ka naam
q 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 hai
p ↔ 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 hain
Isay “p if and only if q” parha jata hai.
25. ~p → ~q kis ka Inverse hai
Original statement p → q ka Inverse banane ke liye dono parts ki negation karte hain: ~p → ~q.
26. q → p kya kehlata hai
Original p → q ke antecedent aur consequent ko switch karne se Converse banta hai: q → p.
27. ~q → ~p kis ka Contrapositive hai
Antecedent aur consequent ko switch karke phir negate karne se Contrapositive banta hai: ~q → ~p.
28. Contrapositive hamesha kya hota hai
Contrapositive apni original statement ke logically equivalent hota hai — matlab dono ka truth value hamesha same rehta hai.
29. Truth table kya hoti hai
Woh table jo compound statement ke tamam possible truth value combinations dikhati hai, usay Truth Table kehte hain.
30. n variables ke liye rows ki tadad
Formula hai 2ⁿ — jitne variables hon, 2 ko unki power tak raise karte hain.
31. 2 variables (p,q) ke liye rows
2² = 4 rows honge.
32. Hamesha true wali statement
Aisi statement jo har condition mein true rahe, usay Tautology kehte hain.
33. Hamesha false wali statement
Aisi statement jo har condition mein false rahe, usay Contradiction kehte hain.
34. Kabhi true kabhi false wali statement
Aisi statement jiska truth value components par depend karta hai (sometimes true, sometimes false), usay Contingency kehte hain.
35. p ∨ ~p
Yeh 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 ∧ ~p
Yeh 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 statements
Do 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 hai
Double negation apni original value pe wapis aa jati hai, isliye ~(~p) = p.
41. (p→q) ∧ (q→p) ka short form
Yeh dono conditionals mila kar Biconditional banta hai: p ↔ q.
42. ∀ symbol ka naam
Isay Universal Quantifier kehte hain.
43. ∃ symbol ka naam
Isay 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 hai
Jab conclusion premises se logically follow kare (chahe premises factually true hon ya na hon), to argument valid kehlata hai.
47. Modus Ponens rule
Agar p → q true hai aur p true hai, to zaroori hai ke q bhi true ho. Yeh rule Modus Ponens kehlata hai.
48. Modus Tollens rule
Agar p → q true hai aur ~q (q ki negation) true hai, to zaroori hai ke ~p bhi true ho. Yeh rule Modus Tollens kehlata hai.
49. “If p then q” ka doosra naam
Isay Implication ya Conditional statement kehte hain.