Question

Is the set `{1, -1}` closed under multiplication and/or addition?

Answer 1

`{1, -1}` is closed with respect to multiplication, but not addition.

Explanation:

Here's a multiplication table for `{1, -1}`...

`underline(color(white)(0+0$)|color(white)(0+)1color(white)(0)color(white)(|0)-1color(white)(0))`
`color(white)(0+)1color(white)(0)|color(white)(0+)1color(white)(0)color(white)(|0)-1color(white)(0)`
`color(white)(0)-1color(white)(0)|color(white)(0)-1color(white)(0)color(white)(|0+)1color(white)(0)`

Regardless of which element we multiply by which, we get an element of the set. So `{1, -1}` is closed under multiplication.

The same cannot be said of addition, since `1 + (-1) = 0` is not in the set `{1, -1}`

`color(white)()`
Footnote

Closure is one of the axioms of a group:

A group is a set `S` equipped with a binary operation `@` satisfying the following properties:

• Closure: If `a, b in S` then `a@b in S`

• Identity: There is an element `I in S` such that `a@I = I@a = a` for any `a in S`

• Inverse: For any `a in S` there is an element `b in S` such that `a@b = b@a = I`

A commutative group also satisfies:

• Commutativity: `a@b = b@a` for all `a, b in S`

The set `S = {1, -1}` with multiplication `@ = xx` satisfies all of these axioms, so is a commutative group.

The normal name of this particular group is `C_2`, the cyclic group of order `2`.