Question

how would i prove that the union of a relation is equivalent to the intersection of ita complement

Answer 1

let us prove this by using an example

let the universal relation consists the all the positive integers.

U={1,2,3,4,5......n}

let Relation 1 R1={1,2,3,4,6}

Relation 2 R2={2,4,5}

we need to prove that union of relation is equivalent to the intersection of its complements.

I.e

R1 U R2= (R1^c ∩ R2^c)^c

R1 U R2= set of all elements present in either R1 or R2

so, R1 U R2={1,2,3,4,5,6}

R1^c= set all elements present in universal set except in R1

so R1^c={5,7,8,9,10,.....n}

similarly R2^c={1,3,6,7,8,9,10.....n}

R1^c∩ R2^c intersection gives the common elements in both the relations.

so R1^c∩ R2^c returns {7,8,9,10.....n}

now we need to find the (R1^c∩ R2^c )^c is the intersection of its complements

(R1^c∩ R2^c )^c = U - (R1^c∩ R2^c )

={1,2,3,4,5,6}

there fore it is proved that (R1 U R2) = (R1^c∩ R2^c )^c