Use Math method to prove those question 1If a b are natural

Use Math method to prove those question

(1)If a; b are natural numbers are a + b and a b natural numbers?

(2)If a; b are integers, are a + b and a b integers?

(3)Please characterize zero in a way which distinguishes it from all other integers using its
properties under addition and multiplication.

(4)Please characterize zero in a way which distinguishes it from all other integers using its
properties under addition and multiplication.

(5)We call an integer a even if there is an integer k such that
a = 2k:
Is 0 an even number? Please prove or disprove

(6)We call an integer a even if there is an integer k such that
a = 2k:
  If a is an even integer and b is any integer is ab even? Please prove or disprove

Solution

1) a,b are natural numbers.

a+b = again a natural number, as natural numbers have closure property under addition.

Similarly ab is a natural number as natural numbers have closure property under multiplication.

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2)

a,b are integers.

a+b = again a integer, as integers have closure property under addition.

Similarly ab is a integer as integers have closure property under multiplication

3)

a+0 =0+a =a

Similarly a(0) = 0(a)=0

These two are the unique properties of 0 which no other integer has.

4) a+0 =0+a =a

Similarly a(0) = 0(a)=0

These two are the unique properties of 0 which no other integer has.

5) Consider 0.

2x0 =0

Thus 0 is a multiple of 2. Hence 0 is even.

6) Given that a is even

So a = 2k for some positive integer k.

ab = 2k(b) = 2(bk)

bk is a positive integer as both b and k are positive.

Hence ab is even.