SOLUTION:-

{{(0,6] ∪ (3,8]} ∩ (1,9)}

=> In interval notation

=> (0,6] => {1, 2, 3, 4, 5, 6} ...(semi open interval)

=> (3,8] => {4, 5, 6, 7, 8} ....(semi open interval)

=> (1,9) => {2, 3, 4, 5, 6, 7, 8) ...(open interval)

now,

{{(0,6] ∪ (3,8]} ∩ (1,9)}

=> {1, 2, 3, 4, 5, 6} ∪ {4, 5, 6, 7, 8} ∩ {2, 3, 4, 5, 6, 7, 8)

=> {1, 2, 3, 4, 5, 6, 7, 8} ∩ {2, 3, 4, 5, 6, 7, 8}

=> {2, 3, 4, 5, 6, 7, 8}

WRITING in interval notation

=> {2, 3, 4, 5, 6, 7, 8} = (1, 8]

Answer is (1, 8]

• INFORMATION about Intervals:

Open Interval: Let a, b ∈ R and a < b then the set {x/x∈R, a < x < b} is called open internal and is denoted by (a,b). All the numbers between a and b BELONGS to the open internal (a,b) but a, b themselves do not belonging to this interval.

Closed Interval: Let a, b ∈R and a < b then the set {x/x ∈R, a <= x <= b} is called closed interval and is denoted by [a, b]. All the numbers between a and b belongs to closed interval [a, b]. Also a and b belongs to this interval.

Semi - closed interval: [a,b) = {x/x ∈R, a <= x < b}

Semi - open interval: (a, b] = {x/x ∈R, a < x <= b}