Q.a) Which one of the following sentences is a STATEMENT?
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Start your 14-day free trial to unlock the full solution →A mathematical statement is a declarative sentence that is definitely true or definitely false; only (i) qualifies. √2 is shown irrational by assuming the contrary and reaching a contradiction.
a) (i) '275 is a perfect square' has a definite truth value (it is false, but still a statement) → this is a STATEMENT. (ii) is an opinion, (iii) is a command, (iv) is not definite, so none of those are statements. Answer: (i).
b) Prove '√2 is irrational' by contradiction.
Assume the opposite: √2 is rational, so √2 = p/q where p, q are integers with no common factor (q ≠ 0).
Then 2 = p²/q² ⇒ p² = 2q², so p² is even ⇒ p is even. Write p = 2m. …
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