Question

已知A={X/2a≤X≤a+1}B={-2≤X≤3}若A∩B=A求实数a取值范围

设全集I=R已知集合M={X/(X+3)的平方≤0}N={X/X的平方+x-6=0}
1，求（CiM)∩N
2，记集合A=（CiM)∩N,已知集合B={X/a-1≤X≤5-a,a属于R},若A∪B=A求a范围

Answer 1

(1)
A= {x| a≤x≤a+1}

B={x| -2≤x≤3}
A∩B=A
=> A is subset of B
=> -2≤a and a+1≤3
=> -2≤a and a≤2
=>-2≤a≤2
(2)
(i)
U =R

M={x| (x+3)^2≤0 }

= {-3}
N={x|x^2+x-6=0 }
={x| (x+3)(x-2)=0 }
={ 2, -3}

Cu(M) = R\{-3}
Cu(M)∩N = R\{-3} ∩{ 2, -3}
={2}

(ii)
A=Cu(M)∩N = {2}
B= { x| a-1≤x≤5-a }

AUB= A
=> A is subset of B

case 1: A=Φ
5-a < a-1
a>3

case 2: A≠Φ ie a≤3
A is subset of B
=> a-1 = 2 and 2=5-a
=> a= 3 and a= 3
solution for case 2 : a= 3

A is subset of B
case 1 or case 2
a>3 or a=3
=> a≥3