Let be the statement ' is irrational' and be the statement ' is irrational'. Which of the following is correct? (Here is a real number.)

Answer and explanation

Correct answer: C. is a necessary but not a sufficient condition for

  1. Step 1: If is rational then is rational, so by contraposition, if is irrational then is irrational: .
  2. Step 2: But is irrational while is rational, so .
Answer
is a necessary but not a sufficient condition for .
Why not the others: is necessary for but not sufficient: irrational numbers can have rational squares.

Practise more Necessary and sufficient conditions.

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