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  • State which of the following statements are true and which are false. Justify your answer. (i) 35 in {x | x has exactly four positive factors}. (ii) 128in {y | the sum of all the positive factors of y is 2y} (iii) 3 ∉ {x | x4 – 5x3 + 2x2 – 112x + 6 = 0}

State which of the following statements are true and which are false. Justify your answer.
(i)    35 \in \{x : x\text{ has exactly four positive factors}\}
(ii)   128 \in \{y :\text{ the sum of all the positive factors of }y\text{ is }2y\}
(iii)   3 \notin \{x : x^4 - 5x^3 + 2x^2 - 112x + 6 = 0\}
(iv)  496 \notin \{y :\text{ the sum of all the positive factors of }y\text{ is }2y\}

Answers (1)

(i)   Given that: 35 \in \{x : x\text{ has exactly four positive factors}\}

        Factors of 35 are 1, 5, 7, 35.

        Hence, true
(ii)   Given that:  128 \in \{y :\text{ the sum of all the positive factors of }y\text{ is }2y\}

        Factors of 128 are 1,2,4,8,16,32,64,128

        sum of all factors 255 \neq 2\times 128

        hence statement is false.
(iii)   Given that: 3 \notin \{x : x^4 - 5x^3 + 2x^2 - 112x + 6 = 0\}

        x^4 - 5x^3 + 2x^2 -112x + 6 = 0

        Put x  = 3

        - 366 = 0 which is not true, So 3 is not an element of the set
        Hence, statement is true
(iv)  Given that: 496 \notin \{y :\text{ the sum of all the positive factors of }y\text{ is }2y\}

        The positive factors of 496 are 1,2,4,8,16, 31, 62, 124, 248, and 496

        The sum of all positive factors is 992 = 2\times 496 which is true, so the 496 is element of the set . Hence the statement is false.

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