Question

Let A be a set of numbers.

• We say that A is closed under addition provided that x + y ∈ A for all x, y ∈ A.
• We say that A is closed under multiplication provided that xy ∈ A for all x, y ∈ A. For each of the following sets, determine whether or not it is closed under addition and whether or not it is closed under multiplication. Justify your answers.

a. The set of all integers divisible by 5.
b. 2n + 1 .
c. {2, 5, 8, 11, 14, . . . }.
d. n ∈ Z.
e. The set of nonzero real numbers.

Answer 1

a. closed under addition and multiplication

b. not closed under addition but closed under multiplication.

c. not closed under addition and multiplication

d. closed under addition and multiplication

e. not closed under addition but closed under multiplication

Explanation:

a.

Let A be a set of all integers divisible by 5.

Let
`x,y`∈A such that
`x=5m\,,\,y=5n`

Find
`x+y,xy`

`x+y=5m+5n=5(m+n)`

So,
`x+y` is divisible by 5.

`xy=(5m)(5n)=25mn=5(5mn)`

So,

`xy` is divisible by 5.

Therefore, A is closed under addition and multiplication.

b.

Let A = 2n +1

Let
`x,y`∈A such that
`x=2m+1\,,\,y=2n+1` where m, n ∈ Z.

Find
`x+y,xy`

`x+y=2m+1+2n+1=2m+2n+2=2(m+n+1)`

So,

`x+y` ∉ A

`xy=(2m+1)(2n+1)=4mn+2m+2n+1=2(2mn+m+n)+1`

So,

`xy`∈ A

Therefore, A is not closed under addition but A is closed under multiplication.

c.

`Let A=\{2,5,8,11,14,...\}`

Let
`x=2,y=5` but
`x+y=2+5=7`∉A

Also,

`xy=2(5)=10`∉A

Therefore, A is not closed under addition and multiplication.

d.

Let A = { 17n: n∈Z}

Let
`x,y` ∈ A such that
`x=17n,y=17m`

Find x + y and xy

`x+y=17n+17m=17(n+m)`

`xy=(17m)(17n)=289mn=17(17mn)`

So,

`x+y,xy` ∈ A

Therefore, A is closed under addition and multiplication.

e.

Let A be the set of nonzero real numbers.

Let
`x,y` ∈ A such that
`x=-1,y=1`

Find x + y

`x+y=-1+1=0`

So,

`x+y` ∈ A

Also, if x and y are two nonzero real numbers then xy is also a non-zero real number.

Therefore, A is not closed under addition but A is closed under multiplication.