Fuzzy Set Union

 Question

        Let  X = {2, 4, 6, … ,20} be the Universe of discourse, 

        `\A^\~ 1 = {\left( 2,0.5 \right),\left( 4,0.9 \right),\left( 6,0.2 \right),\left( 8,0.4 \right),\left( 10,0.5 \right),\left( 12,0.2 \right),\left( 14,0.4} \right),\left( {16,0.0} \right),\left( 18,0.5 \right),\left( 20,0.8 \right)}`

        And 

         `\B^{ \~ 1} = \{ \left( 2,0.5 \right),\left( 4,0.5 \right),\left( 6,0.5 \right),\left( 8,0.2 \right),\left( 10,0.3} \right),\left(12,0.3 \right),\left( 14,0.5 \right),\left( 16,0.8 \right),\left( 18,0.2\right),\left(20,0.4 \right)}`

        Be two fuzzy sets on X. Determine `\A^{ \~ 1} \cup B^{ \~ 1}`

    Solutions:

        Here

            X = {2, 4, 6, … ,20} be the Universe of discourse.

        First, we find `\A^{ \~1}` and `\B^{ \~1}`

        We know that

            `\A^{ \~1} = 1 - \mu A^{ \~ 1}`

        So,

            `\A^{ \~1} = {\left( 2,0.5 \right),\left( 4,0.1 \right),\left( 6,0. \right),\left( 8,0.6 \right),\left( 10,0.5 \right),\left( 12,0.8 \right),\left( 14,0.6 \right),\left( 16,1.0 \right),\left( 18,0.5 \right),\left( 20,0.2 \right)}`

        And 

        `\B^{ \~1} = { \left( 2,0.5 \right),\left( 4,0.5 \right),\left( 6,0.5 \right),\left( 8,0.8 \right),\left( 10,0.7 \right),\left( 12,0.7 \right),\left( 14,0.5 \right),\left( 16,0.2 \right),\left( 18,0.8 \right),\left( 20,0.6 \right)}`

        Now

        `\A^{ \~ 1} \cup B^{ \~ 1`} =  `{\left( 2,0.5 \right),\left( 4,0.5 \right),\left( 6,0.8 \right),\left( 8,0.8 \right),\left( 10,0.7 \right),\left( 12,0.8 \right),\left( 14,0.6 \right),\left( 16,1.0 \right),\left( 18,0.8 \right),\left( 20,0.6 \right)}`