Which set of integers is not closed under what operation?
+
-
x
Division
I thought it was division but I don't understand , can you please explain this.
Thanks
Think of it this way:
If you add two integers, will you always get back an integer ? YES , so it is closed under addition
if you divide two integers will you always get back an integer.
e.g. 6 รท 4 = 1.5 , which is NOT an integer
so not closed under division
make a similar test for subtraction and multiplication.
What do you think?
6*4=24 it is an integer=closed for mult.
10-7=3 so it is also an integer
Thanks
Sure! To understand which set of integers is not closed under division, let's first understand what it means for a set to be closed under an operation.
A set is said to be closed under an operation when performing that operation on any two members of the set always results in another member of the same set.
Now, let's consider each operation and see if the set of integers is closed under it:
1. Addition (+): When you add two integers, you will always get another integer. For example, 3 + 5 = 8, which is also an integer. So, the set of integers is closed under addition.
2. Subtraction (-): Similar to addition, when you subtract two integers, the result is again an integer. For example, 7 - 4 = 3, which is an integer. So, the set of integers is closed under subtraction.
3. Multiplication (x): When you multiply two integers, the result is also an integer. For example, 6 x 2 = 12, which is an integer. So, the set of integers is closed under multiplication.
4. Division (/): Division is different from addition, subtraction, and multiplication. If you divide two integers, the result may not always be an integer. For example, 8 / 3 = 2.6666..., which is not an integer. Therefore, the set of integers is not closed under division.
Hence, the set of integers is not closed under division.