Question

When expressing sets, you may write "inf" for infinity and "U" for union.

Write the complement of the given set in interval notation: (−5,6]

Answer 1

The complement of the interval (-5, 6] would be (-inf, -5] U (6, inf), including all numbers not between -5 and 6, with -5 excluded and 6 excluded.

Step-by-step explanation:

To find the complement of a given set in interval notation, first, consider the entire range of possible values, typically from -infinity to infinity. The complement of a set includes all the numbers not in the original set. The set you provided is (-5, 6]. This interval includes all numbers greater than -5 and up to and including 6, that is, -5 < x ≤ 6.

Therefore, the complement of this set would consist of two parts: all numbers less than or equal to -5, and all numbers greater than 6. Using interval notation and union (U), we express this as:

(-inf, -5] U (6, inf)

Remember that brackets [] are used to indicate that an endpoint is included in the set, known as 'closed', and parentheses () indicate the endpoint is not included, known as 'open'. In this case, -5 is not included in the complement but 6 is excluded as well since 6 is included in the original set.

Answer 2

The complement of the given set in interval notation is
`(-\infty,-5]\cup(6,\infty)`. It can we written as (-inf,5]U(6,inf).

Step-by-step explanation:

The given set in interval notation is

(−5,6]

It means the set is defined as

`A=\x\in R,-5<x\leq 6\`

If B is a set and U is a universal set, then complement of set B contains the elements of universal set but not the elements of set B.

Here, universal set is R, the set set of all real numbers.

`U=\x`

The complement of the given set is

`A^c=U-A`

`A^c=\x\in R,-\infty<x\leq -5,6<x<\infty\`

Complement of the given set in interval notation is

`A^c=(-\infty,-5]\cup(6,\infty)`

Therefore the complement of the given set in interval notation is
`(-\infty,-5]\cup(6,\infty)`. It can we written as (-inf,5]U(6,inf).