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

Q2:

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

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

Q3:

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

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

Q4:

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

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

Q5:

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

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

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

Q9:

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

  • A(𝑋×𝑋)(𝑋×𝑌)
  • B𝑌×(𝑋𝑌)
  • 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.

  • A56
  • B21
  • C77
  • D71
  • E21

Q11:

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

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

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{19,2}
  • B{5,2}
  • C{5,19}
  • D{5,2,19}

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.

  • A𝑅={(2,16),(4,16),(7,9)}
  • B𝑅={(16,2),(16,4),(21,7)}
  • C𝑅={(2,16),(4,16),(7,21)}
  • D𝑅={(2,16),(4,16),(7,21),(2,9)}
  • E𝑅={(2,9),(4,13),(7,16)}

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

Q20:

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

  • A{20}×{37}
  • B{20}×{37,11}
  • C{37,11}×{20}
  • D{20}×{11}

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),(6,2),(6,7),(7,7),(7,2),(2,7),(2,2)}
  • C{(2,2),(2,6),(2,7),(2,2),(6,6),(6,7),(7,2),(7,6)}
  • 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
  • B{(2,3),(3,2)}
  • C{(3,2)}
  • D{(2,3)}

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{(3,7),(3,9),(3,3)}
  • D{(7,3),(9,3),(3,3)}

Q24:

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

  • A{(1,2),(1,5),(3,2),(3,5),(4,2),(4,5)}
  • B{(3,1),(3,3),(3,4),(4,1),(4,3),(4,4)}
  • C{(2,1),(3,1),(4,1),(5,1)}
  • D{(2,3),(2,4),(5,3),(5,4)}
  • 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{(7,8),(7,4),(7,6)}
  • B{(8,7),(4,7),(6,7)}
  • C{(4,7),(6,7),(8,7)}
  • D

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