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how to solve,using the venn diagram?

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2 Answers

Hi Ann,

A Venn diagram shows the relationships between sets.  The Universe is normally drawn as a box, and sets are drawn as circles inside that universe box.  This indicates that each set is included in that universe.  In this case, there are three defined sets, A, B, and C, and they are indeed included in the given universe.

U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19}   Universe: Counting numbers < 20
A = {2,4,6,8,10}  even counting numbers less than 12 (The Counting numbers do not include zero.)
B = {3,6,9,12,15}
C = {5,10,15}

For a good sample of what this might look like, Google "Venn Diagram Purple Math".  Look for a sample drawing with three overlapping circles, then read the rest of this answer to see what to do with it.  (Most of the explanations in the Purple Math article are more advanced than this problem requires.  Just look at the drawings for ideas about how to draw and label the diagram.)

Draw a box and label it "Universe" or something like "Counting Numbers < 20".

Inside the box, draw three overlapping circles and label them A, B, and C.

  1. List the elements in the universe set U as I have done above so you can work with it.
  2. Similarly, list the elements in set A.
  3. Now look for areas where the sets overlap and draw the Venn diagram.
  • We see that sets A and B both include "6".  Write a 6 in the area where circles A and B overlap.
  • Also, sets B and C both include "15", so write 15 in the area where circles B and C overlap.
  • Sets A and C both include "10", so write 10 the area where circles A and C overlap.
  • There are no elements that are in all three sets, so the area where A, B, and C all overlap will be empty.
  • In the part of circle A that does not overlap either B or C, write the remaining elements of set A:
    2, 4, 8.
  • In the part of circle B that does not overlap either A or C, write the remaining element of set B:
    3, 9, 12
  • In the part of circle C that does not overlap either A or B, write the remaining element of set C:
    5
  • Inside the Universe box, but outside all of the circles, write the remaining elements of the universe  (set U) that are not included in any of the sets A, B or C:
    1, 7, 11, 13, 14, 16, 17, 18, 19

You can see that all elements of set A are now inside circle A in the diagram.  The same is true for sets B and C.

That's it.

Hope this helps!

 

A = {2, 4, 6, 8, 10}

B = {3, 6, 9, 12, 15}

C = {5, 10, 15}

A and B  = {6}

A and C = {10}

B and C = {15}

No shared numbers for all three sets.

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