Consider a proposition given as : “ \(x \geq 6\), if \(x^{2} \geq 25\) and its proof as : If \(x \geq 6\), then \(x^{2} = x \cdot x \geq 6 \cdot 6 = 36 \geq 25\) Which of the following is correct w.r.to the given proposition and its proof ? (a) The proof shows the converse of what is to be proved. (b) The proof starts by assuming what is to be shown. (c) The proof is correct and there is nothing wrong.
(a) and (b)
The proposition reads: if \(x^{2} \geq 25\) then \(x \geq 6\).
The proof assumes \(x \geq 6\) and derives \(x^{2} \geq 25\), so it proves the converse; (a) is correct.
Its starting assumption \(x \geq 6\) is exactly the conclusion that had to be shown, so (b) is also correct.
The argument is not valid for the given claim, which is false anyway: \(x = 5\) gives \(x^{2} = 25\) but \(x \lt 6\); so (c) fails.
Hence, (a) and (b).
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