use the venn diagram shown to the right to list the set (b ∪ c) in roster form. (b ∪ c) = (use a comma to…

Explanation:

Step1: Find ( B \cup C )

First, identify all elements in set ( B ) and set ( C ). From the Venn diagram:

• Set ( B ) elements: ( 1, 3, 4, 6, 7, 9, 10, 16, 17 )
• Set ( C ) elements: ( 10, 12, 16, 17, 18, 20 ) The union ( B \cup C ) includes all elements in ( B ) or ( C ), so ( B \cup C={1, 3, 4, 6, 7, 9, 10, 12, 16, 17, 18, 20} )

Step2: Find the complement ( (B \cup C)' )

The universal set ( U ) (from the Venn diagram) has elements: ( 1, 3, 4, 6, 7, 9, 10, 12, 16, 17, 18, 20, ) and the remaining element not in ( B \cup C )? Wait, no, wait—wait, maybe I missed. Wait, let's re - check. Wait, the universal set ( U ) is the rectangle. Let's list all elements in ( U ): looking at the Venn diagram, the elements are ( 1, 3, 4, 6, 7, 9, 10, 12, 16, 17, 18, 20 )? No, wait, no—wait, the element 20 is in ( C )'s outer part, 18 is in ( C )'s outer part, 1 is in ( A )'s outer part, 3 in ( A ), 4 in ( A ), 6 in ( A \cap B ), 7 in ( B ), 9 in ( B ), 10 in ( A \cap C ), 12 in ( C ), 16 in all three, 17 in ( B \cap C ). Wait, no, maybe I made a mistake. Wait, the complement of ( B \cup C ) is the elements in ( U ) but not in ( B \cup C ). Wait, but from the diagram, is there any element not in ( B \cup C )? Wait, no—wait, no, maybe I messed up. Wait, let's re - examine the Venn diagram. Wait, the universal set ( U ) has elements: ( 1, 3, 4, 6, 7, 9, 10, 12, 16, 17, 18, 20 )? No, that can't be. Wait, no, the element that is not in ( B \cup C ): wait, looking at the diagram, the only element not in ( B ) or ( C ) is... Wait, no, wait, maybe I misread. Wait, the set ( A ) has elements ( 1, 3, 4, 6 ) (outer ( A )), ( 6 ) is ( A \cap B ), ( 10 ) is ( A \cap C ), ( 16 ) is ( A \cap B \cap C ). Set ( B ) has ( 6,7,9,16,17 ) (and ( 1 )? No, 1 is in ( A ). Wait, I think I made a mistake earlier. Let's correctly identify the sets:

• Set ( A ): outer ( A ): ( 1, 3, 4 ); ( A \cap B ): ( 6 ); ( A \cap C ): ( 10 ); ( A \cap B \cap C ): ( 16 )
• Set ( B ): outer ( B ): ( 7, 9 ); ( A \cap B ): ( 6 ); ( B \cap C ): ( 17, 16 )
• Set ( C ): outer ( C ): ( 18, 20 ); ( A \cap C ): ( 10 ); ( B \cap C ): ( 17, 16 ); ( C ) only: ( 12 )

So ( B ) elements: ( 6,7,9,16,17 ) (wait, no, ( A \cap B ) is ( 6 ), ( B ) only is ( 7,9 ), ( B \cap C ) is ( 17,16 )) so ( B={6,7,9,16,17} ) ( C ) elements: ( 10,12,16,17,18,20 ) ( ( A \cap C = 10 ), ( C ) only ( 12 ), ( B \cap C=16,17 ), ( C ) only outer ( 18,20 )) So ( B \cup C={6,7,9,10,12,16,17,18,20} ) (wait, no, ( B ) has ( 6,7,9,16,17 ), ( C ) has ( 10,12,16,17,18,20 ), so union is ( 6,7,9,10,12,16,17,18,20 ))

Now, the universal set ( U ) has elements: ( 1,3,4,6,7,9,10,12,16,17,18,20 ) (wait, ( 1,3,4 ) are in ( A ) only). So the complement ( (B \cup C)' ) is the elements in ( U ) but not in ( B \cup C ). So elements in ( U ) are ( 1,3,4,6,7,9,10,12,16,17,18,20 ). Elements in ( B \cup C ) are ( 6,7,9,10,12,16,17,18,20 ). So the elements not in ( B \cup C ) are ( 1,3,4 ).

Wait, that makes sense. Because ( 1,3,4 ) are in ( A ) only, not in ( B ) or ( C ). So ( (B \cup C)'={1,3,4} )

Answer:

( (B \cup C)'={1, 3, 4} )