Question

the diagram shows the number of children who have ever used motorized scooters, manual scooters, and/or bicycles. which combination represents the empty set? motor ∪ manual motor ∪ bicycle motor ∩ manual motor ∩ bicycle

Explanation:

Step1: Recall set - theory definitions

• The union of two sets \(A\cup B=\{x|x\in A\ or\ x\in B\}\).
• The intersection of two sets \(A\cap B = \{x|x\in A\ and\ x\in B\}\). An empty set \(\varnothing\) has no elements.

Step2: Analyze each option

• Option 1: \(motor\cup manual\)
• The set \(motor\cup manual\) contains the elements \(8\), \(24\), and \(32\) (from the Venn - diagram). So, it is not an empty set.
• Option 2: \(motor\cup bicycle\)
• The set \(motor\cup bicycle\) contains the elements \(8\), \(24\), \(44\) (from the Venn - diagram). So, it is not an empty set.
• Option 3: \(motor\cap manual\)
• The set \(motor\cap manual\) contains the element \(24\) (from the Venn - diagram). So, it is not an empty set.
• Option 4: \(motor\cap bicycle\)
• Looking at the Venn - diagram, there is no common region (no overlapping part) between the \(motor\) and \(bicycle\) sets. By the definition of intersection (\(A\cap B=\{x|x\in A\ and\ x\in B\}\)), if there is no common element, the set is empty.

Answer:

motor ∩ bicycle