Worksheet: Cartesian Products

In this worksheet, we will practice performing a Cartesian product and using the operations applied on sets.

Q1:

If 𝑋={8}, 𝑌={8,3}, and 𝑍={9,4,5}, find (𝑋×𝑌)(𝑌×𝑍).

  • A { ( 3 , 5 ) , ( 3 , 8 ) , ( 8 , 9 ) , ( 8 , 8 ) , ( 3 , 9 ) , ( 8 , 5 ) , ( 3 , 4 ) , ( 8 , 4 ) }
  • B { ( 4 , 8 ) , ( 9 , 8 ) , ( 9 , 3 ) , ( 3 , 8 ) , ( 8 , 8 ) , ( 4 , 3 ) , ( 5 , 8 ) , ( 5 , 3 ) }
  • C { ( 8 , 3 ) , ( 8 , 8 ) , ( 3 , 9 ) , ( 3 , 4 ) , ( 3 , 5 ) }
  • D { ( 3 , 5 ) , ( 8 , 3 ) , ( 8 , 9 ) , ( 8 , 8 ) , ( 3 , 9 ) , ( 8 , 5 ) , ( 3 , 4 ) , ( 8 , 4 ) }
  • E { ( 8 , 3 ) , ( 4 , 8 ) , ( 9 , 8 ) , ( 9 , 3 ) , ( 8 , 8 ) , ( 4 , 3 ) , ( 5 , 8 ) , ( 5 , 3 ) }

Q2:

Use the Venn diagram below to find (𝑋𝑌)×𝑌.

  • A { ( 6 , 9 ) , ( 1 , 9 ) , ( 2 , 9 ) , ( 8 , 6 ) , ( 8 , 1 ) , ( 8 , 2 ) }
  • B { ( 9 , 6 ) , ( 9 , 1 ) , ( 9 , 2 ) }
  • C { ( 6 , 9 ) , ( 1 , 9 ) , ( 2 , 9 ) , ( 6 , 8 ) , ( 1 , 8 ) , ( 2 , 8 ) }
  • D { ( 2 , 2 ) , ( 2 , 9 ) , ( 2 , 8 ) , ( 9 , 2 ) , ( 9 , 9 ) , ( 9 , 8 ) }
  • E { ( 9 , 6 ) , ( 9 , 1 ) , ( 9 , 2 ) , ( 6 , 8 ) , ( 1 , 8 ) , ( 2 , 8 ) }

Q3:

If 𝑋={2,3}, 𝑌={5,7}, and 𝑍={7}, find (𝑋×𝑌)(𝑋×𝑍).

  • A { ( 2 , 7 ) , ( 3 , 7 ) }
  • B
  • C { ( 7 , 3 ) , ( 7 , 2 ) }
  • D { ( 2 , 7 ) , ( 3 , 7 ) , ( 2 , 5 ) , ( 3 , 5 ) }

Q4:

If 𝑋={9,5}, 𝑌={3,4}, and 𝑍={4}, find (𝑋𝑌)×𝑍.

  • A { ( 4 , 9 ) , ( 4 , 5 ) }
  • B { ( 4 , 3 ) , ( 4 , 4 ) }
  • C { ( 9 , 4 ) , ( 5 , 4 ) }
  • D { ( 3 , 4 ) , ( 4 , 4 ) }

Q5:

If 𝑋={2}, 𝑌={4}, and 𝑍={2,7}, find (𝑋𝑌)×(𝑌𝑍).

  • A { ( 2 , 4 ) }
  • B { ( 4 , 2 ) }
  • C { ( 2 , 2 ) , ( 7 , 2 ) }
  • D { ( 2 , 2 ) , ( 2 , 7 ) }
  • E { ( 4 , 4 ) }

Q6:

If 𝑋={3,5,7}, 𝑌={9,4,6,7}, and 𝑍={9,4}, find 𝑛(𝑋×(𝑌𝑍)).

Q7:

If 𝑋={8,2}, 𝑌={1,9,4,6}, and 𝑍={4,5}, find (𝑋𝑌)×(𝑍𝑌).

  • A { ( 8 , 1 ) , ( 8 , 4 ) , ( 8 , 5 ) , ( 8 , 6 ) , ( 8 , 9 ) , ( 2 , 1 ) , ( 2 , 4 ) , ( 2 , 5 ) , ( 2 , 6 ) , ( 2 , 9 ) }
  • B { ( 8 , 5 ) , ( 2 , 5 ) }
  • C { ( 8 , 4 ) , ( 2 , 4 ) }
  • D { ( 1 , 4 ) , ( 9 , 4 ) , ( 4 , 4 ) , ( 6 , 4 ) }

Q8:

If 𝑋𝑌={8,3,6}, 𝑌𝑋={4}, and 𝑋𝑌={2}, find (𝑋×𝑌)(𝑌×𝑋).

  • A { ( 8 , 2 ) , ( 3 , 2 ) , ( 6 , 2 ) }
  • B { ( 2 , 2 ) }
  • C { ( 8 , 4 ) , ( 3 , 4 ) , ( 6 , 4 ) }
  • D { ( 4 , 8 ) , ( 4 , 3 ) , ( 4 , 6 ) }

Q9:

If 𝑌={42,22} and 𝑋={7,42}, which of the following is equal to 𝑋×(𝑋𝑌)?

  • A ( 𝑋 × 𝑋 ) ( 𝑋 × 𝑌 )
  • B 𝑌 × ( 𝑋 𝑌 )
  • C ( 𝑋 × 𝑋 ) ( 𝑋 × 𝑌 )
  • D ( 𝑋 × 𝑋 ) ( 𝑋 × 𝑌 )

Q10:

If {31}×{𝑥,𝑦}={(31,25),(31,46)}, find all the possible values of 𝑥+𝑦.

  • A 2 1
  • B77
  • C21
  • D56
  • E71

Q11:

If 𝑋×𝑌={(2,9),(2,6),(7,9),(7,6)}, find 𝑌.

  • A { ( 9 , 9 ) , ( 9 , 7 ) , ( 7 , 9 ) , ( 7 , 7 ) }
  • B { ( 9 , 9 ) , ( 9 , 6 ) , ( 6 , 9 ) , ( 6 , 6 ) }
  • C { ( 2 , 2 ) , ( 2 , 7 ) , ( 7 , 2 ) , ( 7 , 7 ) }
  • D { ( 2 , 2 ) , ( 2 , 6 ) , ( 6 , 2 ) , ( 6 , 6 ) }

Q12:

If 𝑋𝑌 and 𝑋×𝑌={(𝑎,3),(𝑎,2),(𝑎,9),(2,3),(2,2),(2,9)}, find all the possible values of 𝑎.

  • A3 or 9
  • B3 or 2
  • C2
  • D2 or 9

Q13:

If 𝑋={(5,5),(5,2),(5,19),(2,5),(2,2),(2,19),(19,5),(19,2),(19,19)}, find 𝑋.

  • A { 5 , 1 9 }
  • B { 5 , 2 }
  • C { 1 9 , 2 }
  • D { 5 , 2 , 1 9 }

Q14:

If 𝑋×𝑌={(9,8),(9,2),(2,8),(2,2),(6,8),(6,2)}, find 𝑛(𝑋).

Q15:

If 𝑛(𝑋)=2 and 𝑛(𝑌)=13, find 𝑛(𝑋×𝑌).

Q16:

If 𝑋𝑌, 𝑛(𝑋×𝑌)=6, (4,7)𝑋×𝑌, and 5𝑋, then find 𝑌.

  • A { 5 , 4 }
  • B { 4 }
  • C { 6 , 5 , 7 }
  • D { 5 , 4 , 6 }
  • E { 5 , 4 , 7 }

Q17:

For two sets 𝑋 and 𝑌, a function 𝑓 exists from 𝑋 to 𝑌. Also, 𝑎𝑋, 𝑏𝑌, and 𝑎𝑅𝑏 means 𝑏 is divisible by 𝑎. If 𝑋𝑌={2,4,7,9,13,16,21}, 𝑛(𝑋)=3, and 𝑛(𝑋×𝑌)=12, determine 𝑅.

  • A 𝑅 = { ( 2 , 1 6 ) , ( 4 , 1 6 ) , ( 7 , 9 ) }
  • B 𝑅 = { ( 2 , 1 6 ) , ( 4 , 1 6 ) , ( 7 , 2 1 ) }
  • C 𝑅 = { ( 1 6 , 2 ) , ( 1 6 , 4 ) , ( 2 1 , 7 ) }
  • D 𝑅 = { ( 2 , 1 6 ) , ( 4 , 1 6 ) , ( 7 , 2 1 ) , ( 2 , 9 ) }
  • E 𝑅 = { ( 2 , 9 ) , ( 4 , 1 3 ) , ( 7 , 1 6 ) }

Q18:

If 𝑋={9} and 𝑌={2,6}, find 𝑋×𝑌.

  • A { ( 9 , 2 ) , ( 6 , 2 ) }
  • B { ( 2 , 9 ) , ( 6 , 9 ) }
  • C { ( 9 , 2 ) , ( 9 , 6 ) }
  • D { ( 2 , 9 ) , ( 2 , 6 ) }
  • E { ( 6 , 9 ) , ( 6 , 3 ) }

Q19:

If 𝑋×𝑌={(8,9),(8,1),(8,3)}, find 𝑌×𝑋.

  • A { ( 8 , 8 ) }
  • B { ( 9 , 9 ) , ( 9 , 1 ) , ( 9 , 3 ) , ( 1 , 9 ) , ( 1 , 1 ) , ( 1 , 3 ) , ( 3 , 9 ) , ( 3 , 1 ) , ( 3 , 3 ) }
  • C { ( 9 , 8 ) , ( 1 , 8 ) , ( 3 , 8 ) }
  • D { ( 8 , 9 ) , ( 1 , 9 ) , ( 3 , 9 ) }

Q20:

Which of the following Cartesian products would give the result {(20,37),(20,11)}?

  • A { 2 0 } × { 3 7 }
  • B { 3 7 , 1 1 } × { 2 0 }
  • C { 2 0 } × { 1 1 }
  • D { 2 0 } × { 3 7 , 1 1 }

Q21:

If 𝑋={2,6,7}, find 𝑋.

  • A { ( 2 , 2 ) , ( 2 , 6 ) , ( 2 , 7 ) , ( 6 , 2 ) , ( 6 , 7 ) , ( 7 , 7 ) , ( 7 , 2 ) , ( 7 , 6 ) }
  • B { ( 2 , 2 ) , ( 2 , 6 ) , ( 2 , 7 ) , ( 2 , 2 ) , ( 6 , 6 ) , ( 6 , 7 ) , ( 7 , 2 ) , ( 7 , 6 ) }
  • C { ( 2 , 2 ) , ( 2 , 6 ) , ( 2 , 7 ) , ( 6 , 2 ) , ( 6 , 7 ) , ( 7 , 7 ) , ( 7 , 2 ) , ( 2 , 7 ) , ( 2 , 2 ) }
  • D { ( 2 , 2 ) , ( 2 , 6 ) , ( 2 , 7 ) , ( 6 , 2 ) , ( 6 , 6 ) , ( 6 , 7 ) , ( 7 , 2 ) , ( 7 , 6 ) , ( 7 , 7 ) }

Q22:

If 𝑋={2,3}, find 𝑋×.

  • A { ( 3 , 2 ) }
  • B { ( 2 , 3 ) }
  • C
  • D { ( 2 , 3 ) , ( 3 , 2 ) }

Q23:

If 𝑋={9,3,7} and 𝑌={3,4}, find (𝑋×𝑌)𝑌.

  • A { ( 3 , 4 ) , ( 3 , 3 ) }
  • B { ( 3 , 3 ) , ( 4 , 3 ) }
  • C { ( 7 , 3 ) , ( 9 , 3 ) , ( 3 , 3 ) }
  • D { ( 3 , 7 ) , ( 3 , 9 ) , ( 3 , 3 ) }

Q24:

If 𝑋={1}, 𝑌={3,4}, and 𝑍={2,5}, find (𝑍𝑌)×(𝑋𝑌).

  • A { ( 2 , 3 ) , ( 2 , 4 ) , ( 5 , 3 ) , ( 5 , 4 ) }
  • B { ( 3 , 1 ) , ( 3 , 3 ) , ( 3 , 4 ) , ( 4 , 1 ) , ( 4 , 3 ) , ( 4 , 4 ) }
  • C { ( 1 , 2 ) , ( 1 , 5 ) , ( 3 , 2 ) , ( 3 , 5 ) , ( 4 , 2 ) , ( 4 , 5 ) }
  • D { ( 2 , 1 ) , ( 3 , 1 ) , ( 4 , 1 ) , ( 5 , 1 ) }
  • E { ( 2 , 1 ) , ( 2 , 3 ) , ( 2 , 4 ) , ( 5 , 1 ) , ( 5 , 3 ) , ( 5 , 4 ) }

Q25:

If 𝑋={8,4,6}, 𝑌={6,7}, and 𝑍={7}, find 𝑋×(𝑌𝑍).

  • A { ( 4 , 7 ) , ( 6 , 7 ) , ( 8 , 7 ) }
  • B { ( 7 , 8 ) , ( 7 , 4 ) , ( 7 , 6 ) }
  • C { ( 8 , 7 ) , ( 4 , 7 ) , ( 6 , 7 ) }
  • D

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