Question

Set Fn gives all factors of n. Set Mn gives all multiples of n less than 1000. Which of the following statements is/are true?
i. F108 ∩ F84 = F12
ii. M12 ∪ M18 = M36
iii.M12 ∩ M18 = M36
iv.M12 ⊂ M6 ∩ M4
i, ii and iii only
i, iii and iv only
i and iii only
All statements are true

Answer 1

Set Fn gives all factors of n. Set Mn gives all multiples of n less than 1000. Which of the following statements is/are true?

i. F108 ∩ F84 = F12 is the answer

Answer 2

on;y i,iii,iv are true

Explanation:

This is more a question on Number Systems than on Set Theory.

i) F108 ∩ F84 = Set of all numbers that are factors of both 108 and 84 => this is set of all common factors of 84 and 108 => this is set of numbers that are factors of the Highest Common Factor of 84 and 108. HCF (84, 108) = 12. F108 ∩ F84 = F12. Statement A is true.

ii) M12 will have numbers {12, 24, 36, 48, ….} Numbers like 12, 24, … will not feature in M36. So, Statement ii cannot be true.

iii) M12 ∩ M18= Set of all numbers that are multiples of both 12 and 18. => this is set of all common multiples of 12 and 18 => this is set of numbers that are multiples of the least Common Multiple of 12 and 18. LCM (12, 18) = 36. M12 ∩ M18= M36. Statement iii is true.

iv) M6 ⊂ M4 is identical to M12 {Explanation is exactly as we have seen in statement iii}. So, Statement iv is also true. Bear in mind that M12 is a subset of itself. Statement iv is true.