Answer:
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
Step-by-step explanation:
a.
Let A be a set of all integers divisible by 5.
Let ∈A such that
Find
So, is divisible by 5.
So,
is divisible by 5.
Therefore, A is closed under addition and multiplication.
b.
Let A = { 2n +1 | n ∈ Z}
Let ∈A such that
where m, n ∈ Z.
Find
So,
∉ A
So,
∈ A
Therefore, A is not closed under addition but A is closed under multiplication.
c.
Let but
∉A
Also,
∉A
Therefore, A is not closed under addition and multiplication.
d.
Let A = { 17n: n∈Z}
Let ∈ A such that
Find x + y and xy
So,
∈ A
Therefore, A is closed under addition and multiplication.
e.
Let A be the set of nonzero real numbers.
Let ∈ A such that
Find x + y
So,
∈ 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.