Write Which of the Following Statements Are True
Write which of the following statements are true? Justify your answer.
(i) The set of all integers 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 \[ A=\{x:x \text{ is a letter of the word “LITTLE”}\} \] and \[ B=\{x:x \text{ is a letter of the word “TITLE”}\} \] are equal.
Solution
(i) True
Every integer can be written in the form \[ \frac{p}{q} \] where \[ q \neq 0 \] Therefore, every integer is a rational number.
(ii) True
Every crow is a bird. Hence, the set of all crows is contained in the set of all birds.
(iii) False
Every square is a rectangle, but every rectangle is not a square. Therefore, the set of all rectangles is not contained in the set of all squares.
(iv) True
Every real number is also a complex number of the form \[ a+0i \]
(v) False
The set \[ P=\{a\} \] contains the element \[ a \]
But the set \[ B=\{\{a\}\} \] contains the set \[ \{a\} \] as its element.
Therefore, \[ P \neq B \]
(vi) True
The letters of the word “LITTLE” are \[ \{L,I,T,E\} \]
The letters of the word “TITLE” are \[ \{T,I,L,E\} \]
Both sets contain the same elements. Therefore, \[ A=B \]