Question

Please explain the closure property in the following set:
1.)The set of natural numbers is closed under addition.
2.)The set of prime numbers is closed under multiplication.
3.)The odd numbers is closed under multiplication.

Answer 1

1
2
3

Answer 2

PLEASE answer this one.i can't do this

Answer 3

1) If you add two positive integers (natural numbers) you get a positive integer, so the natural numbers are closed under addition. Proving this is NOT trivial at all and requires a set-theoretical construction of the natural numbers, but it should be intuitive at least.
2) If you multiply two prime numbers a and b, you never get a prime number since the new number will have a and b as divisors.
3) If you multiply two odd numbers you always get an odd number:\[(2n+1)(2m+1) = 2(2mn + m + n) + 1\]so the odd numbers are closed under multiplication.

Answer 4

can u give me some examples for number 1 and number 2?

Answer 5

1) Adding any two natural numbers yields another natural number:\[2 + 2 = 4\]\[5 + 6 = 11\]\[34 + 23 = 57\]etc.
2) Multiplying any two prime numbers always yield a compositive (that is, not prime) number:
\[2 \cdot 2 = 4\]and 4 is divisible by 2, so it's composite;
\[5 \cdot 2011 = 10055\]and 10055 is divisible by 5 and 2001, so it's composite;
\[2 \cdot 97 = 194\]and 194 is by 2 and 97, so it's composite; etc.

Answer 6

thank u so much!