# Closure

Closure is when an operation (such as "adding") on members of a set (such as "real numbers") always makes a member of the same set. |

So the result stays in the same set.

### Example: when we add two real numbers we get another real number

3.1 + 0.5 = 3.6

This is always true, so: **real numbers are closed under addition**

### Example: subtracting two whole numbers might **not** make a whole number

4 − 9 = −5

−5 is **not** a whole number (whole numbers can't be negative)

So: **whole numbers are not closed under subtraction**

This is a **general idea**, and can apply to any sort of operation on any kind of set!

### Example: the set of shirts

For the operation "wash", the shirt is still a shirt after washing.

- So shirts are
**closed**under the operation "wash"

For the operation "rip", a small rip may be OK, but a shirt ripped in half ceases to be a shirt!

- So shirts are
**not closed**under the operation "rip"

## Sets

A set is a collection of things (usually numbers). Examples:

- Set of even numbers: {..., -4, -2, 0, 2, 4, ...}
- Set of odd numbers: {..., -3, -1, 1, 3, ...}
- Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
- Positive multiples of 3 that are less than 10: {3, 6, 9}

Let's look more closely at one set:

### Example: Odd numbers {..., -3, -1, 1, 3, ...}

Is the set of odd numbers closed under the simple operations + − × ÷ ?

- Adding? 3 + 7 = 10 but 10 is even, not odd, so
**no** - Subtracting? 11 − 3 = 8 but 8 is even, not odd, so
**no** - Multiplying? 5 × 7 = 35 yes ... in fact multiplying odd numbers
**always**produces odd numbers, so**odd numbers are closed under multiplication** - Dividing? 33/3 = 11 which looks good! But try 33/5 = 6.6 which is not odd, so
**no**

As we just saw, just one case where it does **NOT** work is enough to say it is **NOT** closed.

But to say it **IS** closed, we must know it is **ALWAYS** closed (just one example could fool us).