Are the following sets closed? Explain why / why not?
a) The set {1, 2, 3, 4, 5} under addition?
b) The set {-1, 0, 1} under addition?
c) The integers under addition?
d) The set of positive integers under subtraction?
e) The set of natural numbers under division?
a) yes
b) no: 1+1 is not an element of the set
what do you think of the others?
oops
a) no: any sum bigger than 5 is not in the set
Let's analyze each set and determine if they are closed under the given operation.
a) The set {1, 2, 3, 4, 5} under addition:
To check if this set is closed under addition, we need to verify if the sum of any two elements in the set is also in the set. In this case, for any two numbers chosen from the set, say x and y, let's check if x + y is in the set.
For example, if we take x = 2 and y = 5, the sum of x + y is 7, which is not in the set. Therefore, the set {1, 2, 3, 4, 5} is not closed under addition.
b) The set {-1, 0, 1} under addition:
Similarly, we need to check if the sum of any two elements in the set is also in the set. Let's take x = -1 and y = 1. Here, x + y = 0, which is in the set {-1, 0, 1}. Therefore, the set {-1, 0, 1} is closed under addition.
c) The integers under addition:
To check if the integers are closed under addition, we need to verify if the sum of any two integers is also an integer. For any two integers x and y, x + y will always be an integer since addition is a closed operation over the set of integers. Therefore, the integers are closed under addition.
d) The set of positive integers under subtraction:
For this set to be closed under subtraction, the result of subtracting any two positive integers should also be a positive integer. Let's take x = 5 and y = 8. Here, x - y = 5 - 8 = -3, which is not a positive integer. Therefore, the set of positive integers is not closed under subtraction.
e) The set of natural numbers under division:
For this set to be closed under division, the result of dividing any two natural numbers should also be a natural number. Let's take x = 6 and y = 2. Here, x / y = 6 / 2 = 3, which is a natural number. Therefore, the set of natural numbers is closed under division.
In conclusion, the sets that are closed under the given operations are:
b) The set {-1, 0, 1} under addition.
c) The integers under addition.
e) The set of natural numbers under division.