Question

Which of the following statements are true? Justify your answers.

i) The collection of all Class 5 students is a set.

ii) {«} is the empty set.

iii) No purely imaginary number is a real number.

iv) For any z∈C, if |z| , then ∈R z∈ R .

v) A biquadratic equation must have at least one real root.

vi) The system of equations 2x + 3y + z5 = 1and 5 3x + 2y + z = 5has a unique solution.

vii) The AM of 1,0 − ,1 is greater than or equal to their GM.

viii) ‘A is sufficient for B’ is equivalent to ‘the negative of A is necessary for the
negative of B’.

ix) The Gaussian elimination method can only be applied for solving a system of 4
linear equations if the number of variables is 4.

Answer 1

1. True. All Class 5 students is a well-defined collection with clear distinguishing property.
2. False. This set contains one element, so it is not empty.
3. True. All purely imaginary numbers contain imaginary unit so can not be real.
4. False. Counterexample: $z = 1+i\in C$ , despite $|z| = \sqrt{2} \in R\\$ .
5. False. Any biquadratic equation can be reduced to a quadratic equation. And not any quadratic equation has real roots and so biquadratic one.
6. False. It is the system with 2 equations and 3 unknowns. It always has infinetly many solutions.
7. True. $AM = (-1 +0+1)/3 = 0, GM = \sqrt[3]{(-1)\times0\times1}=0$
8. True. ‘A is sufficient for B’ can be written as: $\neg A \lor B$ . ‘The negative of A is necessary for the negative of B’ can be written as: $\neg (\neg B) \lor \neg A$ , which is equivalent to the previous statement.
9. False. The Gaussian elimination method can be applied to the system with any number of equations and variables.