If a = {x : 1 < x < 9} ; b = {x : 0 < x < 9} ; C = {x : 2 x < 9} and a, b, c U, what are the elements in U?

The question has lost some of its symbols in formatting (for example, "C = {x : 2 x < 9}" is missing a sign, and the inequality signs may originally have been ≤). So let's reason from what we have.

Step 1: Understand the question

The universal set U must contain every element of A, B and C. So U must be at least as large as the union of the three sets: A ∪ B ∪ C ⊆ U.

Step 2: Read the sets

• A = {x : 1 < x < 9} (the original likely used ≤ in places)
• B = {x : 0 < x < 9}
• C = {x : 2 ≤ x < 9}

The set B has the widest range, starting just above 0 and going up to below 9. A and C lie inside B:

• A (from 1 to 9) lies inside B (from 0 to 9)
• C (from 2 to 9) lies inside B

So A ∪ B ∪ C = B = {0 < x < 9}.

Step 3: Compare with the options

• A. {-1 < x < 9}: contains everything, but is larger than needed.
• B. {0 < x < 9}: exactly matches the union, and contains A, B and C.
• C. {1 < x < 9}: leaves out the values between 0 and 1 that are in B, so B is not a subset. Not valid.
• D. {1 < x < 10}: also misses the part of B between 0 and 1. Not valid.
• E. {2 < x < 10}: misses much of A and B. Not valid.

Option A would technically contain the sets too, but the intended answer is the smallest set that contains all three, which is the one matching the widest set, B.

Answer: B, U = {0 < x < 9}.

Tip: When asked for the universal set of several given sets, find the union of all of them. Look for the smallest lower bound and the largest upper bound, and choose the option that matches that range.