Question

To list the16 different relations on the set \((0,1\} \).

Short answer

The 16 different relations on the set\(\{ 0,1\} \)is given below:

\(\begin{array}{l}{R_1} = \phi \\{R_2} = \{ (0,0)\} \\{R_3} = \{ (0,1)\} \\{R_4} = \{ (1,0)\} \\{R_5} = \{ (1,1)\} \\{R_6} = \{ (0,0),(0,1)\} \\{R_7} = \{ (0,0),(1,0)\} \\{R_8} = \{ (0,0),(1,1)\} \\{R_9} = \{ (0,1),(1,0)\} \\{R_{10}} = \{ (0,1),(1,1)\} \\{R_{11}} = \{ (1,0),(1,1)\} \\{R_{12}} = \{ (0,0),(0,1),(1,0)\} \\{R_{13}} = \{ (0,0),(0,1),(1,1)\} \\{R_{14}} = \{ (0,0),(1,0),(1,1)\} \\{R_{15}} = \{ (0,1),(1,0),(1,1)\} \\{R_{16}} = \{ (0,0),(0,1),(1,0),(1,1)\} \end{array}\)

Step-by-step solution

 Given

The set \(\{ 0,1\} \) and need to list 16 different relations.

The Concept of relation

A relation from a set A to a set B is a subset of A×B. Hence, a relation R consists of ordered pairs (a,b), where a∈A and b∈B.

Determine the relation

The set\(\{ 0,1\} \)and need to list 16 different relations.

Let us consider\(A = \{ 0,1\} \). Then,\(A \times A = \{ (0,0),(0,1),(1,0),(1,1)\} \). So, the subsets of\(A \times A\)are precisely the relations on\(A\).

Therefore, the relation on\(A = \{ 0,1\} \)are:

\(\begin{array}{l}{R_1} = \phi \\{R_2} = \{ (0,0)\} \\{R_3} = \{ (0,1)\} \\{R_4} = \{ (1,0)\} \\{R_5} = \{ (1,1)\} \\{R_6} = \{ (0,0),(0,1)\} \\{R_7} = \{ (0,0),(1,0)\} \\{R_8} = \{ (0,0),(1,1)\} \\{R_9} = \{ (0,1),(1,0)\} \\{R_{10}} = \{ (0,1),(1,1)\} \\{R_{11}} = \{ (1,0),(1,1)\} \\{R_{12}} = \{ (0,0),(0,1),(1,0)\} \\{R_{13}} = \{ (0,0),(0,1),(1,1)\} \\{R_{14}} = \{ (0,0),(1,0),(1,1)\} \\{R_{15}} = \{ (0,1),(1,0),(1,1)\} \\{R_{16}} = \{ (0,0),(0,1),(1,0),(1,1)\} \end{array}\)

Hence, it is shown.

Conclusion:

Hence the 16 different relations on the set \(\{ 0,1\} \) is as shown.