The set of all perfect numbers and twin primes among the numbers 1, 2, 3, 4, 5, 7, 6 , 7 , 8 , 9 is:

Given:

The set of all perfect numbers and twin primes among the numbers {1, 2, 3, 4, 5, 7, 6, 7, 8, 9}.

Formula used:

A perfect number is a positive integer equal to the sum of its proper divisors (excluding itself).

A pair of twin primes are two prime numbers that differ by 2.

Calculation:

Numbers given: {1, 2, 3, 4, 5, 7, 6, 7, 8, 9}

Identify perfect numbers:

- For 6: Proper divisors = {1, 2, 3}. Sum = 1 + 2 + 3 = 6. Hence, 6 is a perfect number.
- No other number in the set satisfies this property.

Identify twin primes:

- Twin primes from the set: (3, 5) and (5, 7).

Combine perfect numbers and twin primes:

- Perfect number: {6}

- Twin primes: {3, 5, 7}

Combined set = {3, 5, 6, 7}

∴ The correct answer is option (4).