Question

What is the complement of B relative to A for sets A = `[1,5,8,10,12]` and B = `2,4,5,8,11]`?

I know how to find the complement of B, but I don't understand what relative to A really means.

Thanks

Answer 1

See below.

Explanation:

Our given sets are:

`A={1,5,8,10,12}` and `B={2,4,5,8,11}`

The complement of `B` relative to `A` usually expressed as `A\\\B` literally means every thing in `A` that is not in `B`.

Notice in the example we have an intersection:

`AnnB={5,8}`

These elements belong to both `A` and `B`, but because they are not exclusively in `A` they do not form part of the relative complement.

The notation `A\\\B` is a generalization of subtraction, `A-B`, where we are subtracting the elements from `A` that are also in `B`.

We could see this as:

`color(white)(8888888888888.)A-B`
`{1,cancel(5),cancel(8),10,12}-{2,4,5,8,11}`

`{1,10,12}`

So the complement of `B` relative to `A` is:

`A\\\ B={1,10,12}`

This is represented by the shaded area in the Venn diagram below:

`color(white)(88888888888888)`Complement of A relative to B ( A\B )

The complement of `B` expressed `B'` is everything that is not in `B`, but this includes the universal space as well ( the area inside the rectangle usually denoted by `U`. This is represented by the shaded region in the Venn diagram below, and as you can see includes `U`. This is sometimes referred to as the absolute complement.

`color(white)(888888888888888888)`Absolute complement of B'