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:

  • A𝑅={(8,6),(12,10),(13,11)}
  • B𝑅={(6,8),(10,13),(11,12)}
  • C𝑅={(6,8),(10,12),(11,13),(8,6),(12,10),(13,11)}
  • D𝑅={(6,8),(10,12),(13,11)}
  • E𝑅={(6,8),(10,12),(11,13)}

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 Made0123
Total Points0369
  • 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.

  • A𝑅={(2,4),(4,6),(8,10),}
  • B𝑅={(2,4),(4,10),(8,6),}
  • C𝑅={(2,4),(4,6),(10,8),}
  • D𝑅={(4,2),(6,4),(10,8),}
  • E𝑅={(2,4),(4,6),(8,10),(4,2),(6,4),(10,8),}

Q4:

Write the relation 𝑅 for the following arrow diagram:

  • A𝑅={(9,9),(8,8)}
  • B𝑅=9,19,8,18,(0,0),8,18,9,19
  • C𝑅={(9,9),(8,8),(8,8),(9,9)}
  • D𝑅={9,8,0}
  • E𝑅={(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.

  • A𝑏=𝑎+2
  • B𝑏=35𝑎
  • C5𝑎=3𝑏
  • D𝑏=𝑎2
  • E𝑎+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?

  • A𝑏=𝑎+1
  • B𝑏=2𝑎2
  • C𝑏=2𝑎+2
  • D𝑎=2𝑏2
  • E𝑎=2𝑏+2

Q7:

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

  • A𝑅={(18,18),(9,9),(0,0),(9,9),(18,18)}
  • B𝑅=18,118,9,19,(0,0),9,19,18,118
  • C𝑅={(18,18),(9,9)}
  • D𝑅={(18,18),(9,9),(9,9),(18,18)}
  • E𝑅={18,9,0,9,18}

Q8:

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

  • A𝑅={(4,4),(7,7),(4,6),(5,8),(6,5),(7,8)}
  • B𝑅={(4,4),(7,7),(8,8),(6,4),(8,5),(5,6),(8,7)}
  • C𝑅={(4,4),(7,7),(8,8),(4,6),(5,8),(6,5),(7,8)}
  • D𝑅={(4,4),(7,7),(8,8),(8,5),(5,6),(8,7)}
  • E𝑅={(4,4),(8,8),(4,6),(5,8),(6,5),(7,8)}

Q9:

Write the relation 𝑅 for the following arrow diagram:

  • A𝑅={(11,6),(14,9),(15,10)}
  • B𝑅={(6,11),(9,15),(10,14)}
  • C𝑅={(6,11),(9,14),(10,15),(11,6),(14,9),(15,10)}
  • D𝑅={(6,11),(9,14),(15,10)}
  • E𝑅={(6,11),(9,14),(10,15)}

Q10:

Write the relation 𝑅 for the following arrow diagram:

  • A𝑅={(1,5),(4,2),(6,0)}
  • B𝑅={(5,1),(2,6),(0,4)}
  • C𝑅={(5,1),(2,4),(0,6),(1,5),(4,2),(6,0)}
  • D𝑅={(5,1),(2,4),(6,0)}
  • E𝑅={(5,1),(2,4),(0,6)}

Q11:

Write the relation 𝑅 for the following arrow diagram:

  • A𝑅={(3,8),(2,7),(1,6)}
  • B𝑅={(8,3),(7,1),(6,2)}
  • C𝑅={(8,3),(7,2),(6,1),(3,8),(2,7),(1,6)}
  • D𝑅={(8,3),(7,2),(1,6)}
  • E𝑅={(8,3),(7,2),(6,1)}

Q12:

Write the relation 𝑅 for the following arrow diagram:

  • A𝑅={(2,1),(5,2),(8,5)}
  • B𝑅={(1,2),(2,8),(5,5)}
  • C𝑅={(1,2),(2,5),(5,8),(2,1),(5,2),(8,5)}
  • D𝑅={(1,2),(2,5),(8,5)}
  • E𝑅={(1,2),(2,5),(5,8)}

Q13:

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

  • A𝑅={(5,10),(10,15),(20,25),}
  • B𝑅={(5,10),(10,25),(20,15),}
  • C𝑅={(5,10),(10,15),(25,20),}
  • D𝑅={(10,5),(15,10),(25,20),}
  • E𝑅={(5,10),(10,15),(20,25),(10,5),(15,10),(25,20),}

Q14:

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

  • A𝑅={(4,8),(8,12),(16,20),}
  • B𝑅={(4,8),(8,20),(16,12),}
  • C𝑅={(4,8),(8,12),(20,16),}
  • D𝑅={(8,4),(12,8),(20,16),}
  • E𝑅={(4,8),(8,12),(16,20),(8,4),(12,8),(20,16),}

Q15:

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

  • A𝑅={(3,6),(6,9),(12,15),}
  • B𝑅={(3,6),(6,15),(12,9),}
  • C𝑅={(3,6),(6,9),(15,12),}
  • D𝑅={(6,3),(9,6),(15,12),}
  • E𝑅={(3,6),(6,9),(12,15),(6,3),(9,6),(15,12),}

Q16:

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

  • A𝑅={(6,12),(12,18),(24,30),}
  • B𝑅={(6,12),(12,30),(24,18),}
  • C𝑅={(6,12),(12,18),(30,24),}
  • D𝑅={(12,6),(18,12),(30,24),}
  • E𝑅={(6,12),(12,18),(24,30),(12,6),(18,12),(30,24),}

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.

  • A𝑅={(8,11),(7,9),(10,15)}
  • B𝑅={(8,13),(7,12),(10,15)}
  • C𝑅={(8,21),(7,19),(10,25)}
  • D𝑅={(21,8),(19,7),(25,10)}

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={9,6,49,25}, and 𝑅 is a relation from 𝑋 to 𝑌, where 𝑎𝑅𝑏 means that 𝑎=𝑏 for each 𝑎𝑋 and 𝑏𝑌. Determine the relation 𝑅.

  • A𝑅={(3,9),(7,49),(5,25)}
  • B𝑅={(3,9),(7,49),(5,6)}
  • C𝑅={(9,3),(49,7),(25,5)}
  • D𝑅={(3,49),(7,9),(5,25)}

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.

  • A𝑅={(25,5),(0,0),(25,5)}
  • B𝑅={(5,25),(0,0),(5,25)}
  • C𝑅={(25,5),(25,5)}
  • D𝑅={(5,25),(5,25)}

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.

  • A𝑅={(9,3),(1,3),(4,3),(6,3),(9,2),(1,2),(4,2),(6,2),(1,7),(4,7),(6,7)}
  • B𝑅={(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)}
  • C𝑅={(3,9),(3,1),(3,4),(3,6),(2,9),(2,1),(2,4),(2,6),(7,1),(7,4),(7,6)}
  • D𝑅={(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.

  • A𝑅={(20,20),(20,1),(20,3)}
  • B𝑅={(20,20),(20,1),(20,3),(1,20),(1,3),(3,20),(3,1),(3,3)}
  • C𝑅={(20,20),(20,1),(20,3),(1,20),(1,1),(1,3),(3,20),(3,1),(3,3)}
  • D𝑅={(20,20),(20,1)}

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.

  • A𝑅={(1,3),(3,1)}
  • B𝑅={(1,1),(1,2),(1,3),(2,1),(2,2)}
  • C𝑅={(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}
  • D𝑅={(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.

  • A𝑥=5, and 𝑦=2
  • B𝑥=4, and 𝑦=2 or 𝑦=2
  • C𝑥=4, and 𝑦=6 or 𝑦=6
  • D𝑥=4, and 𝑦=2

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