Q:

Let S = {a, b, c). Find the following:a) the number of reflexive relations on Sb)the number of reflexive and symmetric relations on S

Accepted Solution

A:
Answer:The number of reflexive relations on S  is 64.The number of reflexive and symmetric relations on S  is 8.Step-by-step explanation:Consider the provided set S = {a, b, c}.The number of elements in the provided set is 3.Part (a) the number of reflexive relations on STo calculate the number of reflexive relation on S we can use the formula as shown:Total number of Reflexive Relations on a set: [tex]2^{n(n-1)}[/tex].Where, n is the number of elements.In the provided set we have 3 elements, so substitute the value of n in the above formula:[tex]2^{3(3-1)}[/tex][tex]2^{3(2)}[/tex][tex]2^{6}[/tex][tex]64[/tex]Hence, the number of reflexive relations on S  is 64.Part(b) The number of reflexive and symmetric relations on S.To calculate the number of reflexive and symmetric relation on S we can use the formula as shown:Total number of Reflexive and symmetric Relations on a set: [tex]2^{\frac{n(n-1)}{2}}[/tex].Where, n is the number of elements.In the provided set we have 3 elements, so substitute the value of n in the above formula:[tex]2^{\frac{3(3-1)}{2}}[/tex][tex]2^{\frac{3(2)}{2}}[/tex][tex]2^{\frac{6}{2}}[/tex][tex]2^{3}[/tex][tex]8[/tex]Hence, the number of reflexive and symmetric relations on S  is 8.