Question

Write which of the following statements are true ? Justify your answer.

(i) The set of all intergers is contained in the set of all rational numbers.

(ii) The set of all crows is contained in the set of all birds.

(iii) The set of all rectangles is contained in the set of all squares.

(iv) The set of all real numbers is contained in the set of all complex numbers.

(v) The sets P = {a} and B = {{a}} are equal.

(vi) The sets A = {x : x is a letter of the word "LITTLE"} and B = {x : x is a letter of the word "TITLE"} are equal.

Answer 1

(i) The given statement is 'True'.

If m $ϵ$ z, then m can be written as $\frac{m}{1}$, which is of the form $\frac{p}{q}$, where p and q are relatively prime intergers and q $\ne$ 0.

This implies that $mϵQ$, the set rational numbers.

Thus, $mϵZ\Rightarrow mϵQ$

Hence $Z\subseteq Q.$

(ii) The given statement is 'True'.

$\because$ Crows are also Birds.

(iii) The given statement is 'False'.

$\because$ A rectangle need not be a square.

(iv) The given statement is 'True'.

If z is a complex numbers, then it can be written as z = x +iy,

where x and y ar real numbers and are called the real and imaginary parts of the complex number z.

If x is a real number, then

x = x +i, $0ϵC,$

where C is the set of complex numbers.

Thus $xϵR\Rightarrow xϵC.$

Hence, the set of all real numbers is contained in the set of all complex numbers.

(v) False, $\because aϵPbuta\text{/}ϵB$

Note that {a} is an element of B which is different from the elements 'a'.

(vi) A = {L, I, T, E}

[$\because$ reprtition is not allowed]

B = {T, I, L, E}

[$\because$ reprtition is not allowed]

= {L, I, T, E}

[$\because$ the manner in which the elements are listed does not matter ]

$\because$ Each element of A is an element of B and vice-versa

$\therefore$ A = B

Hence, the given statement is true.