Consider a proposition given as: \(𝑥 ≥ 6\), if \(𝑥^2≥25\) and and its proof…
2015
Consider a proposition given as:
\(𝑥 ≥ 6\), if \(𝑥^2≥25\) and and its proof as:
If \(𝑥 ≥ 6\), then \(𝑥^2=𝑥.𝑥≥6.6=36≥25\)
Which of the following is correct with respect to the given proposition and its proof?
(a) The proof shows the converse
(b) The proof starts by assuming what is to be shown
(c) The proof is correct and there is nothing wrong
- A.
(a) only
- B.
(c) only
- C.
(a) and (b)
- D.
(b) only
Attempted by 108 students.
Show answer & explanation
Correct answer: C
Interpretation of the original proposition: The phrase "x ≥ 6 if x^2 ≥ 25" means "If x^2 ≥ 25 then x ≥ 6."
What the given proof does: The proof begins by assuming x ≥ 6 and then deduces x^2 = x·x ≥ 6·6 = 36 ≥ 25. That proves the implication "If x ≥ 6 then x^2 ≥ 25," which is the converse of the intended implication. Because the proof assumes x ≥ 6 (the conclusion of the original statement), it is also starting by assuming what it was supposed to prove.
Correctness of the original statement: The original implication "If x^2 ≥ 25 then x ≥ 6" is not generally true. For example, x = −5 gives x^2 = 25 but x ≥ 6 is false. Thus the proof neither proves the intended claim nor provides a valid argument for it.
Key points:
The proof shows the converse: it proves "If x ≥ 6 then x^2 ≥ 25."
The proof starts by assuming the conclusion (x ≥ 6), so it assumes what it was supposed to prove.
Therefore the correct selection is the choice that asserts both "the proof shows the converse" and "the proof starts by assuming what is to be shown."