Mathematical Reasoning • Topic 3 of 3

Quantifiers

Quantifiers say how many. The universal quantifier ∀ ("for all", "every", "each") claims something holds for every member of a set; the existential quantifier ∃ ("there exists", "some", "at least one") claims it holds for at least one. The crucial CAT skill is negation, because most "which must be true" traps hinge on it. The negation of "for all x, P(x)" is "there exists an x for which P(x) is false" — one counterexample destroys a universal claim. The negation of "there exists an x with P(x)" is "for all x, P(x) is false". A frequent error is over-negating: the opposite of "all are red" is "at least one is not red", NOT "none is red". Order matters too: "every student has a mentor" (∀ then ∃) is weaker than "there is a mentor for every student" (∃ then ∀). Read the quantifier, find the scope, and to disprove a universal claim just hunt for a single counterexample.

✅ Solved examples

1. Negate: "Every prime number is odd."
Negation of ∀ is ∃ with the predicate negated: "There exists a prime number that is not odd." (True — 2 is the counterexample.)
2. Negate: "Some students failed the mock."
Negation of ∃ is ∀ negated: "No student failed the mock" i.e. "All students passed the mock."
3. Is "All multiples of 6 are even" true, and what would disprove it?
True. A counterexample would be a multiple of 6 that is odd; none exists, since every multiple of 6 is also a multiple of 2, so the universal claim stands.
4. Distinguish "Every lock has a key" from "There is a key for every lock."
The first is ∀lock ∃key (each lock has its own key — weaker). The second, read literally as ∃key ∀lock, claims one master key opens all locks — a stronger statement. The quantifier order changes the meaning.

✏️ Practice — try these, take hints as needed

1. Negate: "All CAT aspirants take a mock test."
Universal → existential.
Negate the predicate.
One counterexample suffices.
At least one CAT aspirant does not take a mock test.
2. Negate: "There exists a student who scored 100 percentile."
Existential → universal.
Negate inside.
No such student.
No student scored 100 percentile (every student scored below 100).
3. True or false: the negation of "all are red" is "none is red".
Negation of ∀ is ∃ ¬.
It should be "at least one is not red".
"None is red" is too strong.
False — the correct negation is "at least one is not red".
4. Negate: "Some politicians are honest."
Existential statement.
Becomes a universal.
Negate the predicate.
No politician is honest (all politicians are not honest).
5. Does "Every number greater than 2 is composite" hold? Give the disproof tool.
Look for a prime above 2.
A counterexample disproves a universal.
3 is prime.
False — 3 (a prime above 2) is a counterexample.

📝 Topic test — 8 questions

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