Statement & conclusion: Reasoning Ability Notes for Nagaland Patwari / Lekhpal

Study Notes – Statement & Conclusion

(Reasoning Ability – IIT‑JEE / Competitive Exams)

Aim:
• Understand the basic logic behind Statement and Conclusion questions.
• Learn the systematic procedure to deduce conclusions that are true, false or cannot be decided from the given statement.
• Master the use of “If‑Then” relations, “Only‑If”, “Either‑or”, “Not‑This‑but‑That” etc.
• Become comfortable with logical notations and truth‑tables for quick evaluation.

1. Definition

Term	Meaning	Logical Symbol
Statement (S)	A complete sentence that is either True or False.	“S”
Conclusion (C)	A statement that follows from the given statement(s).	“C”
True Conclusion	C is necessarily true when S is true.
False Conclusion	C is necessarily false when S is true.
Undetermined (NEITHER)	C cannot be deduced; may be true or false.

2. Basic Logical Operators

Symbol	English	Example	Truth‑Table
¬	NOT	¬S = “S is not true.”	T → F, F → T
∧	AND	S ∧ C	T∧T = T, others = F
∨	OR	S ∨ C	T∨T = T, T∨F = T, F∨F = F
→	IF…THEN	S → C	T→T = T, T→F = F, F→T = T, F→F = T
↔	IF AND ONLY IF	S ↔ C	T↔T = T, F↔F = T, others = F

Note: In Statement–Conclusion questions we rarely use ↔ explicitly; it is hidden in the wording.

3. Common Patterns

Pattern	Logical Form	Example	Verdict
“If … then …”	A → B	“If it rains, then the ground is wet.”	A is S, B is C
“Only if”	B → A	“The ground is wet only if it rains.”	Equivalent to “If wet, then rain.”
“Either … or …”	A ∨ B	“Either it rains or it is sunny.”	Both may be true; cannot decide.
“Neither … nor …”	¬A ∧ ¬B	“Neither the cat nor the dog is in the house.”	Two negatives.
“Not only … but also …”	A ∧ B	“The exam is not only difficult but also interesting.”	Both true.
“If … but …”	A → B (but C)	“If he studies, then he will pass, but he did not study.”	Contradiction – leads to C = False.
“If … then … and …”	(A → B) ∧ (A → C)	“If it is Monday, then the market is open and the school is closed.”	Two separate conclusions.

4. Systematic Procedure

Read the Statement Carefully.
Identify the premise(s) and any explicit conditions or exceptions.
Translate into Logical Form.
Use the operators above.
Example:
Statement: “If the train is delayed, then Mr. R will miss the meeting.” → D → M.
List All Possible Interpretations.
For a single conditional, there are 4 combinations:
- D = True, M = True → S true, C true.
- D = True, M = False → S true, C false.
- D = False, M = True → S false, C true.
- D = False, M = False → S false, C false.
Apply the Statement’s Truth (S = True).
- If the statement is true (as given), then only those rows where S = True survive.
- Discard the rest.
Check the Conclusion.
- If all surviving rows give the same truth value for C → “True” or “False” conclusion.
- If the values vary → “Undetermined (NEITHER).”
Remember “Contrapositive”
(A → B) is equivalent to (\neg B → \neg A).
Helpful for If…but… type statements.

5. Derivation Tricks

Trick	When to Use	Example
Contradiction	If the statement is “If A then B” and the conclusion is “not B”.	“If it rains, the ground is wet.” Conclusion: “The ground is not wet.” → False.
Reductio ad Absurdum	Assume conclusion true and show it violates statement.	“If the teacher is absent, the class is cancelled.” Conclusion: “The class is cancelled.” → True.
Transitive Property	Combine two statements: A → B and B → C → A → C.	“If it rains, the ground is wet.” “If ground is wet, students stay indoors.” → “If it rains, students stay indoors.”
Exclusive OR	“Either A or B but not both.”	“Either the book is on the table or on the shelf.” → Cannot decide.
Double Negation	Simplify ¬¬A → A.	“It is not the case that the door is not open.” → Door is open.

6. Common Pitfalls

Pitfall	Why It Happens	How to Avoid
Assuming universality	Thinking “If A then B” implies “If not A then not B.”	Remember only the forward implication holds.
Confusing “Only if” with “If”