Question

If R is a binary relation on a set A define R^–1 on A.

Let R = {(a, b) : a, b ϵ W and 3a + 2b = 15} and 3a + 2b = 15}, where W is the set of whole numbers.

Express R and R^–1 as sets of ordered pairs.

Show that (i) dom (R) = range (R^–1) (ii) range (R) = dom (R^–1)

Answer 1

3a + 2b = 15

b = \(\frac{15-3a}{2}\)

a=1 è b=6 

a=3 è b=3 

a=5 è b=0

R = {(1, 6), (3, 3), (5, 0)}

R⁻¹ = {(6, 1), (3, 3), (0, 5)} 

The domain of R is the set of first co-ordinates of R 

Dom(R) = {1, 3, 5} 

The range of R is the set of second co-ordinates of R 

Range(R) = {6, 3, 0} 

The domain of R⁻¹ is the set of first co-ordinates of R⁻¹ 

Dom(R ⁻¹) = {6, 3, 0} 

The range of R⁻¹ is the set of second co-ordinates of R⁻¹ Range(R⁻¹) = {1, 3, 5} 

Thus,

dom (R) = range (R^–1) 

range (R) = dom (R^–1)