Problem 12 · 2021 Math Kangaroo
Hard
Algebra & Patterns
substitution
If \(A = {]0,1[} \cup {]2,3[}\) and \(B = {]1,2[} \cup {]3,4[}\), what is the set of all numbers of the form \(a+b\) with \(a \in A\) and \(b \in B\)? (Here \(]m,n[\) denotes the open interval from m to n.)
Show answer
Answer: D — \(]1,3[ \cup ]3,5[ \cup ]5,7[\)
Show hints
Hint 1 of 2
Add each piece of A to each piece of B; adding two open intervals gives another open interval.
Still stuck? Show hint 2 →
Hint 2 of 2
Take the union of all the resulting intervals.
Show solution
Approach: add the intervals pairwise and union the results
- A = (0,1)∪(2,3), B = (1,2)∪(3,4). Adding endpoints: (0,1)+(1,2)=(1,3); (0,1)+(3,4)=(3,5); (2,3)+(1,2)=(3,5); (2,3)+(3,4)=(5,7).
- The union is (1,3) ∪ (3,5) ∪ (5,7) — note 3 and 5 are never reached.
- That matches ]1,3[ ∪ ]3,5[ ∪ ]5,7[.
Mark:
· log in to save