Here are the elements of each set based on the given definitions:
The universal set is U={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}.
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A={x:x∈Uandxiseven}
A={2,4,6,8,10,12,14,16,18}
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B={x:x∈Uandxislessthan10}
B={1,2,3,4,5,6,7,8,9}
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C={x:x∈Uandxisamultipleof3}
C={3,6,9,12,15,18}
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D={x:x∈Uandxisaprimenumber}
D={2,3,5,7,11,13,17}
Now, let's solve each part:
a) A∪B′
Step 1: Find B′.
B′=U−B={10,11,12,13,14,15,16,17,18}
Step 2: Find A∪B′.
A∪B′={2,4,6,8,10,12,14,16,18}∪{10,11,12,13,14,15,16,17,18}
A∪B′={2,4,6,8,10,11,12,13,14,15,16,17,18}
b) A∩C∩D
Step 1: Find A∩C.
A∩C={2,4,6,8,10,12,14,16,18}∩{3,6,9,12,15,18}
A∩C={6,12,18}
Step 2: Find (A∩C)∩D.
(A∩C)∩D={6,12,18}∩{2,3,5,7,11,13,17}
(A∩C)∩D=∅
c) B∩(C∪D)′
Step 1: Find C∪D.
C∪D={3,6,9,12,15,18}∪{2,3,5,7,11,13,17}
C∪D={2,3,5,6,7,9,11,12,13,15,17,18}
Step 2: Find (C∪D)′.
(C∪D)′=U−(C∪D)={1,4,8,10,14,16}
Step 3: Find B∩(C∪D)′.
B∩(C∪D)′={1,2,3,4,5,6,7,8,9}∩{1,4,8,10,14,16}
B∩(C∪D)′={1,4,8}
d) (A∪B′)−B
Step 1: From part a), A∪B′={2,4,6,8,10,11,12,13,14,15,16,17,18}.
Step 2: Find (A∪B′)−B.
(A∪B′)−B={2,4,6,8,10,11,12,13,14,15,16,17,18}−{1,2,3,4,5,6,7,8,9}
(A∪B′)−B={10,11,12,13,14,15,16,17,18}
e) (A−B)′
Step 1: Find A−B.
A−B={2,4,6,8,10,12,14,16,18}−{1,2,3,4,5,6,7,8,9}
A−B={10,12,14,16,18}
Step 2: Find (A−B)′.
(A−B)′=U−(A−B)={1,2,3,4,5,6,7,8,9,11,13,15,17}
(A−B)′={1,2,3,4,5,6,7,8,9,11,13,15,17}
f) ∣C∪D∣
Step 1: From part c), C∪D={2,3,5,6,7,9,11,12,13,15,17,18}.
Step 2: Count the number of elements in C∪D.
∣C∪D∣=12
g) (A−B)′∩A
Step 1: From part e), (A−B)′={1,2,3,4,5,6,7,8,9,11,13,15,17}.
Step 2: Find (A−B)′∩A.
(A−B)′∩A={1,2,3,4,5,6,7,8,9,11,13,15,17}∩{2,4,6,8,10,12,14,16,18}
