Mathematical Reasoning • Topic 1 of 3

Logical Statements

A statement (or proposition) is a sentence that is definitely either true or false — never both, never neither. "12 is divisible by 4" is a statement; "Solve this!" and "Is it raining?" are not. The negation ¬p flips the truth value: the negation of "all CAT toppers study daily" is NOT "no CAT topper studies daily" but the weaker "at least one CAT topper does not study daily". Compound statements join simple ones with connectives: AND (∧) demands both parts, OR (∨) is inclusive and demands at least one. The CAT-relevant skill is reading everyday English into logic precisely — "but", "yet" and "although" are all AND; "unless" usually means "if not". When you negate a compound statement use De Morgan: ¬(p ∧ q) becomes ¬p ∨ ¬q, and ¬(p ∨ q) becomes ¬p ∧ ¬q. Getting the negation right is what separates a valid inference from a trap answer.

✅ Solved examples

1. Which is a statement: (a) "x + 2 = 5" with x unknown, (b) "Mumbai is in India", (c) "Please be quiet"?
Only (b) is definitely true or false, so (b) is a statement. (a) is an open sentence (depends on x); (c) is a command.
2. Negate: "Ravi is tall and thin."
By De Morgan, ¬(p ∧ q) = ¬p ∨ ¬q. So: "Ravi is not tall OR Ravi is not thin."
3. Negate: "He will pass or he will repeat the year."
¬(p ∨ q) = ¬p ∧ ¬q. So: "He will not pass AND he will not repeat the year."
4. Write "She studies hard but does not relax" using connectives, then negate it.
"But" = AND, so it is p ∧ ¬q where p = studies hard, q = relaxes. Negation = ¬p ∨ q = "She does not study hard OR she relaxes."

✏️ Practice — try these, take hints as needed

1. Is "This sentence is false" a statement?
A statement must be definitely true or false.
Test: assume it is true, then what?
It contradicts itself either way.
No — it is a paradox, neither true nor false.
2. Negate: "The team is strong and well-funded."
It is p ∧ q.
Apply De Morgan.
¬(p ∧ q) = ¬p ∨ ¬q.
The team is not strong OR not well-funded.
3. Negate: "He takes the bus or the metro."
It is p ∨ q.
De Morgan flips OR to AND.
¬(p ∨ q) = ¬p ∧ ¬q.
He does not take the bus AND does not take the metro.
4. Express "She will attend unless it rains" as an implication.
"Unless" means "if not".
Let r = it rains, a = she attends.
If not r then a.
¬r → a (if it does not rain, she attends).
5. Negate: "All managers are punctual."
This is a "for all" statement.
Its negation is "there exists".
One counterexample suffices.
At least one manager is not punctual.

📝 Topic test — 8 questions

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