Worksheet: Representing Relations

In this worksheet, we will practice representing a relation using a mapping diagram or a graph, knowing the domain and range of a given relation.

Q1:

Write the relation 𝑅 for the following arrow diagram:

  • A 𝑅 = { ( 8 , 6 ) , ( 1 2 , 1 0 ) , ( 1 3 , 1 1 ) }
  • B 𝑅 = { ( 6 , 8 ) , ( 1 0 , 1 3 ) , ( 1 1 , 1 2 ) }
  • C 𝑅 = { ( 6 , 8 ) , ( 1 0 , 1 2 ) , ( 1 1 , 1 3 ) , ( 8 , 6 ) , ( 1 2 , 1 0 ) , ( 1 3 , 1 1 ) }
  • D 𝑅 = { ( 6 , 8 ) , ( 1 0 , 1 2 ) , ( 1 3 , 1 1 ) }
  • E 𝑅 = { ( 6 , 8 ) , ( 1 0 , 1 2 ) , ( 1 1 , 1 3 ) }

Q2:

In basketball, each shot made from outside the 3-point line scores 3 points. The table shows this relationship. List this information as ordered pairs (3-point shots made, total number of points).

3-Point Shots Made 0 1 2 3
Total Points 0 3 6 9
  • A ( 1 , 0 ) , ( 3 , 2 ) , ( 3 , 0 ) , ( 9 , 6 )
  • B ( 0 , 0 ) , ( 1 , 0 ) , ( 3 , 2 ) , ( 3 , 0 )
  • C ( 0 , 0 ) , ( 3 , 1 ) , ( 6 , 2 ) , ( 9 , 3 )
  • D ( 0 , 1 ) , ( 2 , 3 ) , ( 0 , 3 ) , ( 6 , 9 )
  • E ( 0 , 0 ) , ( 1 , 3 ) , ( 2 , 6 ) , ( 3 , 9 )

Q3:

Using the Cartesian diagram below, determine the relation 𝑅, where 𝑅𝑥𝑦.

  • A 𝑅 = { ( 2 , 4 ) , ( 4 , 6 ) , ( 8 , 1 0 ) , }
  • B 𝑅 = { ( 2 , 4 ) , ( 4 , 1 0 ) , ( 8 , 6 ) , }
  • C 𝑅 = { ( 2 , 4 ) , ( 4 , 6 ) , ( 1 0 , 8 ) , }
  • D 𝑅 = { ( 4 , 2 ) , ( 6 , 4 ) , ( 1 0 , 8 ) , }
  • E 𝑅 = { ( 2 , 4 ) , ( 4 , 6 ) , ( 8 , 1 0 ) , ( 4 , 2 ) , ( 6 , 4 ) , ( 1 0 , 8 ) , }

Q4:

Write the relation 𝑅 for the following arrow diagram:

  • A 𝑅 = { ( 9 , 9 ) , ( 8 , 8 ) }
  • B 𝑅 = 9 , 1 9 , 8 , 1 8 , ( 0 , 0 ) , 8 , 1 8 , 9 , 1 9
  • C 𝑅 = { ( 9 , 9 ) , ( 8 , 8 ) , ( 8 , 8 ) , ( 9 , 9 ) }
  • D 𝑅 = { 9 , 8 , 0 }
  • E 𝑅 = { ( 9 , 9 ) , ( 8 , 8 ) , ( 0 , 0 ) , ( 8 , 8 ) , ( 9 , 9 ) }

Q5:

Given that 𝑎𝑥 and 𝑏𝑦, express the following arrow diagram in the form of an equation.

  • A 𝑏 = 𝑎 + 2
  • B 𝑏 = 3 5 𝑎
  • C 5 𝑎 = 3 𝑏
  • D 𝑏 = 𝑎 2
  • E 𝑎 + 𝑏 = 2

Q6:

Given that 𝑅 is a relation from 𝑋 to 𝑌, where 𝑎𝑋 and 𝑏𝑌, which of the following equations correctly expresses relation 𝑅?

  • A 𝑏 = 𝑎 + 1
  • B 𝑏 = 2 𝑎 2
  • C 𝑏 = 2 𝑎 + 2
  • D 𝑎 = 2 𝑏 2
  • E 𝑎 = 2 𝑏 + 2

Q7:

Which of the following correctly express the relation 𝑅 illustrated in the figure below?

  • A 𝑅 = { ( 1 8 , 1 8 ) , ( 9 , 9 ) , ( 0 , 0 ) , ( 9 , 9 ) , ( 1 8 , 1 8 ) }
  • B 𝑅 = 1 8 , 1 1 8 , 9 , 1 9 , ( 0 , 0 ) , 9 , 1 9 , 1 8 , 1 1 8
  • C 𝑅 = { ( 1 8 , 1 8 ) , ( 9 , 9 ) }
  • D 𝑅 = { ( 1 8 , 1 8 ) , ( 9 , 9 ) , ( 9 , 9 ) , ( 1 8 , 1 8 ) }
  • E 𝑅 = { 1 8 , 9 , 0 , 9 , 1 8 }

Q8:

Which of the following relations correctly express the mapping diagram shown in the figure below?

  • A 𝑅 = { ( 4 , 4 ) , ( 7 , 7 ) , ( 4 , 6 ) , ( 5 , 8 ) , ( 6 , 5 ) , ( 7 , 8 ) }
  • B 𝑅 = { ( 4 , 4 ) , ( 7 , 7 ) , ( 8 , 8 ) , ( 6 , 4 ) , ( 8 , 5 ) , ( 5 , 6 ) , ( 8 , 7 ) }
  • C 𝑅 = { ( 4 , 4 ) , ( 7 , 7 ) , ( 8 , 8 ) , ( 4 , 6 ) , ( 5 , 8 ) , ( 6 , 5 ) , ( 7 , 8 ) }
  • D 𝑅 = { ( 4 , 4 ) , ( 7 , 7 ) , ( 8 , 8 ) , ( 8 , 5 ) , ( 5 , 6 ) , ( 8 , 7 ) }
  • E 𝑅 = { ( 4 , 4 ) , ( 8 , 8 ) , ( 4 , 6 ) , ( 5 , 8 ) , ( 6 , 5 ) , ( 7 , 8 ) }

Q9:

Write the relation 𝑅 for the following arrow diagram:

  • A 𝑅 = { ( 1 1 , 6 ) , ( 1 4 , 9 ) , ( 1 5 , 1 0 ) }
  • B 𝑅 = { ( 6 , 1 1 ) , ( 9 , 1 5 ) , ( 1 0 , 1 4 ) }
  • C 𝑅 = { ( 6 , 1 1 ) , ( 9 , 1 4 ) , ( 1 0 , 1 5 ) , ( 1 1 , 6 ) , ( 1 4 , 9 ) , ( 1 5 , 1 0 ) }
  • D 𝑅 = { ( 6 , 1 1 ) , ( 9 , 1 4 ) , ( 1 5 , 1 0 ) }
  • E 𝑅 = { ( 6 , 1 1 ) , ( 9 , 1 4 ) , ( 1 0 , 1 5 ) }

Q10:

Write the relation 𝑅 for the following arrow diagram:

  • A 𝑅 = { ( 1 , 5 ) , ( 4 , 2 ) , ( 6 , 0 ) }
  • B 𝑅 = { ( 5 , 1 ) , ( 2 , 6 ) , ( 0 , 4 ) }
  • C 𝑅 = { ( 5 , 1 ) , ( 2 , 4 ) , ( 0 , 6 ) , ( 1 , 5 ) , ( 4 , 2 ) , ( 6 , 0 ) }
  • D 𝑅 = { ( 5 , 1 ) , ( 2 , 4 ) , ( 6 , 0 ) }
  • E 𝑅 = { ( 5 , 1 ) , ( 2 , 4 ) , ( 0 , 6 ) }

Q11:

Write the relation 𝑅 for the following arrow diagram:

  • A 𝑅 = { ( 3 , 8 ) , ( 2 , 7 ) , ( 1 , 6 ) }
  • B 𝑅 = { ( 8 , 3 ) , ( 7 , 1 ) , ( 6 , 2 ) }
  • C 𝑅 = { ( 8 , 3 ) , ( 7 , 2 ) , ( 6 , 1 ) , ( 3 , 8 ) , ( 2 , 7 ) , ( 1 , 6 ) }
  • D 𝑅 = { ( 8 , 3 ) , ( 7 , 2 ) , ( 1 , 6 ) }
  • E 𝑅 = { ( 8 , 3 ) , ( 7 , 2 ) , ( 6 , 1 ) }

Q12:

Write the relation 𝑅 for the following arrow diagram:

  • A 𝑅 = { ( 2 , 1 ) , ( 5 , 2 ) , ( 8 , 5 ) }
  • B 𝑅 = { ( 1 , 2 ) , ( 2 , 8 ) , ( 5 , 5 ) }
  • C 𝑅 = { ( 1 , 2 ) , ( 2 , 5 ) , ( 5 , 8 ) , ( 2 , 1 ) , ( 5 , 2 ) , ( 8 , 5 ) }
  • D 𝑅 = { ( 1 , 2 ) , ( 2 , 5 ) , ( 8 , 5 ) }
  • E 𝑅 = { ( 1 , 2 ) , ( 2 , 5 ) , ( 5 , 8 ) }

Q13:

Using the Cartesian diagram below, determine the relation 𝑅, where 𝑅𝑥𝑦.

  • A 𝑅 = { ( 5 , 1 0 ) , ( 1 0 , 1 5 ) , ( 2 0 , 2 5 ) , }
  • B 𝑅 = { ( 5 , 1 0 ) , ( 1 0 , 2 5 ) , ( 2 0 , 1 5 ) , }
  • C 𝑅 = { ( 5 , 1 0 ) , ( 1 0 , 1 5 ) , ( 2 5 , 2 0 ) , }
  • D 𝑅 = { ( 1 0 , 5 ) , ( 1 5 , 1 0 ) , ( 2 5 , 2 0 ) , }
  • E 𝑅 = { ( 5 , 1 0 ) , ( 1 0 , 1 5 ) , ( 2 0 , 2 5 ) , ( 1 0 , 5 ) , ( 1 5 , 1 0 ) , ( 2 5 , 2 0 ) , }

Q14:

Using the Cartesian diagram below, determine the relation 𝑅, where 𝑅𝑥𝑦.

  • A 𝑅 = { ( 4 , 8 ) , ( 8 , 1 2 ) , ( 1 6 , 2 0 ) , }
  • B 𝑅 = { ( 4 , 8 ) , ( 8 , 2 0 ) , ( 1 6 , 1 2 ) , }
  • C 𝑅 = { ( 4 , 8 ) , ( 8 , 1 2 ) , ( 2 0 , 1 6 ) , }
  • D 𝑅 = { ( 8 , 4 ) , ( 1 2 , 8 ) , ( 2 0 , 1 6 ) , }
  • E 𝑅 = { ( 4 , 8 ) , ( 8 , 1 2 ) , ( 1 6 , 2 0 ) , ( 8 , 4 ) , ( 1 2 , 8 ) , ( 2 0 , 1 6 ) , }

Q15:

Using the Cartesian diagram below, determine the relation 𝑅, where 𝑅𝑥𝑦.

  • A 𝑅 = { ( 3 , 6 ) , ( 6 , 9 ) , ( 1 2 , 1 5 ) , }
  • B 𝑅 = { ( 3 , 6 ) , ( 6 , 1 5 ) , ( 1 2 , 9 ) , }
  • C 𝑅 = { ( 3 , 6 ) , ( 6 , 9 ) , ( 1 5 , 1 2 ) , }
  • D 𝑅 = { ( 6 , 3 ) , ( 9 , 6 ) , ( 1 5 , 1 2 ) , }
  • E 𝑅 = { ( 3 , 6 ) , ( 6 , 9 ) , ( 1 2 , 1 5 ) , ( 6 , 3 ) , ( 9 , 6 ) , ( 1 5 , 1 2 ) , }

Q16:

Using the Cartesian diagram below, determine the relation 𝑅, where 𝑅𝑥𝑦.

  • A 𝑅 = { ( 6 , 1 2 ) , ( 1 2 , 1 8 ) , ( 2 4 , 3 0 ) , }
  • B 𝑅 = { ( 6 , 1 2 ) , ( 1 2 , 3 0 ) , ( 2 4 , 1 8 ) , }
  • C 𝑅 = { ( 6 , 1 2 ) , ( 1 2 , 1 8 ) , ( 3 0 , 2 4 ) , }
  • D 𝑅 = { ( 1 2 , 6 ) , ( 1 8 , 1 2 ) , ( 3 0 , 2 4 ) , }
  • E 𝑅 = { ( 6 , 1 2 ) , ( 1 2 , 1 8 ) , ( 2 4 , 3 0 ) , ( 1 2 , 6 ) , ( 1 8 , 1 2 ) , ( 3 0 , 2 4 ) , }

Q17:

Which of the following all satisfy the relation 𝑦=4?

  • A ( 4 , 4 ) , ( 3 , 4 ) , ( 2 , 4 )
  • B ( 4 , 4 ) , ( 3 , 4 ) , ( 2 , 4 )
  • C ( 4 , 4 ) , ( 4 , 3 ) , ( 4 , 2 )
  • D ( 4 , 4 ) , ( 5 , 5 ) , ( 6 , 6 )

Q18:

Given that 𝑋={8,7,10}, 𝑌={𝑑𝑑}, and 𝑅 is a relation from 𝑋 to 𝑌, where 𝑎𝑅𝑏 means that 𝑏=2𝑎+5 for each 𝑎𝑋 and 𝑏𝑌, determine the relation 𝑅.

  • A 𝑅 = { ( 8 , 1 1 ) , ( 7 , 9 ) , ( 1 0 , 1 5 ) }
  • B 𝑅 = { ( 8 , 1 3 ) , ( 7 , 1 2 ) , ( 1 0 , 1 5 ) }
  • C 𝑅 = { ( 8 , 2 1 ) , ( 7 , 1 9 ) , ( 1 0 , 2 5 ) }
  • D 𝑅 = { ( 2 1 , 8 ) , ( 1 9 , 7 ) , ( 2 5 , 1 0 ) }

Q19:

Given that 𝑋={1,7,4}, 𝑌={3,6,2}, and 𝑅={(1,6),(4,3),(2,2)}, determine whether 𝑅 represents a relation from 𝑋 to 𝑌 or not.

  • Ano
  • Byes

Q20:

𝑋 = { 3 , 7 , 5 } , 𝑌 = { 9 , 6 , 4 9 , 2 5 } , and 𝑅 is a relation from 𝑋 to 𝑌, where 𝑎𝑅𝑏 means that 𝑎=𝑏 for each 𝑎𝑋 and 𝑏𝑌. Determine the relation 𝑅.

  • A 𝑅 = { ( 3 , 9 ) , ( 7 , 4 9 ) , ( 5 , 2 5 ) }
  • B 𝑅 = { ( 3 , 9 ) , ( 7 , 4 9 ) , ( 5 , 6 ) }
  • C 𝑅 = { ( 9 , 3 ) , ( 4 9 , 7 ) , ( 2 5 , 5 ) }
  • D 𝑅 = { ( 3 , 4 9 ) , ( 7 , 9 ) , ( 5 , 2 5 ) }

Q21:

Given that 𝑋={6,5,0,5,6}, 𝑌=[0,36), and 𝑅 is a relation from 𝑋 to 𝑌, where 𝑎𝑅𝑏 means 𝑎=𝑏 for each 𝑎𝑋 and 𝑏𝑌, determine the relation 𝑅.

  • A 𝑅 = { ( 2 5 , 5 ) , ( 0 , 0 ) , ( 2 5 , 5 ) }
  • B 𝑅 = { ( 5 , 2 5 ) , ( 0 , 0 ) , ( 5 , 2 5 ) }
  • C 𝑅 = { ( 2 5 , 5 ) , ( 2 5 , 5 ) }
  • D 𝑅 = { ( 5 , 2 5 ) , ( 5 , 2 5 ) }

Q22:

Given that 𝑋={3,2,8,7}, 𝑌={9,1,4,6}, and 𝑅 is a relation from 𝑋 to 𝑌, where 𝑎𝑅𝑏 means that 𝑎+𝑏<14 for each 𝑎𝑋 and 𝑏𝑌, determine the relation 𝑅.

  • A 𝑅 = { ( 9 , 3 ) , ( 1 , 3 ) , ( 4 , 3 ) , ( 6 , 3 ) , ( 9 , 2 ) , ( 1 , 2 ) , ( 4 , 2 ) , ( 6 , 2 ) , ( 1 , 7 ) , ( 4 , 7 ) , ( 6 , 7 ) }
  • B 𝑅 = { ( 3 , 9 ) , ( 3 , 1 ) , ( 3 , 4 ) , ( 3 , 6 ) , ( 2 , 9 ) , ( 2 , 1 ) , ( 2 , 4 ) , ( 2 , 6 ) , ( 8 , 1 ) , ( 8 , 4 ) , ( 7 , 1 ) , ( 7 , 4 ) , ( 7 , 6 ) }
  • C 𝑅 = { ( 3 , 9 ) , ( 3 , 1 ) , ( 3 , 4 ) , ( 3 , 6 ) , ( 2 , 9 ) , ( 2 , 1 ) , ( 2 , 4 ) , ( 2 , 6 ) , ( 7 , 1 ) , ( 7 , 4 ) , ( 7 , 6 ) }
  • D 𝑅 = { ( 9 , 3 ) , ( 1 , 3 ) , ( 4 , 3 ) , ( 6 , 3 ) , ( 9 , 2 ) , ( 1 , 2 ) , ( 4 , 2 ) , ( 6 , 2 ) , ( 1 , 8 ) , ( 4 , 8 ) , ( 1 , 7 ) , ( 4 , 7 ) , ( 6 , 7 ) }

Q23:

Given that 𝑋={20,1,3}, and 𝑅 is a relation on 𝑋, where 𝑎𝑅𝑏 means that 𝑎+2𝑏 is equal to an even number for each 𝑎𝑋 and 𝑏𝑋, determine the relation 𝑅.

  • A 𝑅 = { ( 2 0 , 2 0 ) , ( 2 0 , 1 ) , ( 2 0 , 3 ) }
  • B 𝑅 = { ( 2 0 , 2 0 ) , ( 2 0 , 1 ) , ( 2 0 , 3 ) , ( 1 , 2 0 ) , ( 1 , 3 ) , ( 3 , 2 0 ) , ( 3 , 1 ) , ( 3 , 3 ) }
  • C 𝑅 = { ( 2 0 , 2 0 ) , ( 2 0 , 1 ) , ( 2 0 , 3 ) , ( 1 , 2 0 ) , ( 1 , 1 ) , ( 1 , 3 ) , ( 3 , 2 0 ) , ( 3 , 1 ) , ( 3 , 3 ) }
  • D 𝑅 = { ( 2 0 , 2 0 ) , ( 2 0 , 1 ) }

Q24:

If 𝑋={𝑥𝑥,1𝑥3}, and 𝑅 is a relation on 𝑋, where 𝑎𝑅𝑏 means 𝑎+𝑏 is divisible by 2 for each 𝑎𝑋 and 𝑏𝑋, determine the relation 𝑅.

  • A 𝑅 = { ( 1 , 3 ) , ( 3 , 1 ) }
  • B 𝑅 = { ( 1 , 1 ) , ( 1 , 2 ) , ( 1 , 3 ) , ( 2 , 1 ) , ( 2 , 2 ) }
  • C 𝑅 = { ( 1 , 1 ) , ( 1 , 2 ) , ( 1 , 3 ) , ( 2 , 1 ) , ( 2 , 2 ) , ( 2 , 3 ) , ( 3 , 1 ) , ( 3 , 2 ) , ( 3 , 3 ) }
  • D 𝑅 = { ( 1 , 1 ) , ( 1 , 3 ) , ( 2 , 2 ) , ( 3 , 1 ) , ( 3 , 3 ) }

Q25:

The relation 𝑅 on natural numbers is defined by 𝑎𝑅𝑏 if and only if 𝑎×𝑏=12. Find 𝑥 such that 𝑥𝑅3, and 𝑦 such that 𝑦𝑅3𝑦.

  • A 𝑥 = 5 , and 𝑦=2
  • B 𝑥 = 4 , and 𝑦=2 or 𝑦=2
  • C 𝑥 = 4 , and 𝑦=6 or 𝑦=6
  • D 𝑥 = 4 , and 𝑦=2

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