# Set Theory, Calendars, Clocks and Binomial Theorem

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## Set Theory: Union and Intersection

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. ${F_{108} \cap F_{84}=F_{12}}$
ii. ${M_{12} \cup M_{18}=M_{36}}$
iii. ${M_{12} \cap M_{18}=M_{36}}$
iv. ${M_{12} \subset (M_{6}\cap M_{4})}$

1. i, ii and iii only
2. i, iii and iv only
3. i and iii only
4. All statements are true

Choice B. i, iii and iv only

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## Detailed Solution

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

i. ${F_{108} \cap F_{84}}$
This is the 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.

${F_{108} \cap F_{84}=F_{12}}$ - this is true.

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

iii. ${M_{12} \cap M_{18}}$ - this is the set of all common multiples of 12 and 18.
This is the set of numbers that are multiples of the Least Common Multiple of 12 and 18. (which is 36).

This statement is also true. iv. Using the same logic as that used in statement iii, we can determine that statement iv is also true. Remember that every set is a subset of itself.

Correct Answer: i, iii and iv are true.

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