Question

# If $U{\text{ = }}\left\{ {a,b,c,d,e,f,g,h} \right\},$ find complements of the following sets. $\left( i \right){\text{ A = }}\left\{ {a,b,c} \right\} \\ \left( {ii} \right){\text{ }}B = {\text{ }}\left\{ {d,e,f,g} \right\} \\ \left( {iii} \right){\text{ C = }}\left\{ {a,c,e,g} \right\} \\ \left( {iv} \right){\text{ D = }}\left\{ {f,g,h,a} \right\} \\$

Hint – In order to solve this question, we have to find out the complements of the given sets presented in the universal set that is represented by $U$. For set A we have to find out the elements not in A. Similarly, we have to find out the complements of the sets B,C,D in the universal set

Universal set- A universal set is a collection of all objects in a particular theory. All other sets in that framework constitute subsets of the universal sets which is denoted as an uppercase Italic letter $\left( U \right)$ .

For complement $\left( {A'} \right)$
$\left( {A'} \right) = U - A$ (Universal set – elements present in A)
=$\left\{ {d,e,f,g,h} \right\}$

For complement $\left( {B'} \right)$
$\left( {B'} \right) = U - B{\text{ }} \\ = \left\{ {a.b,c,h} \right\} \\$

For complement $\left( {C'} \right)$
$\left( {C'} \right) = U - C \\ = \left\{ {b,d,f,h} \right\} \\$

For complement $\left( {D'} \right)$
$\left( {D'} \right) = U - D \\ = \left\{ {b,c,d,e} \right\} \\$

Note – In order to solve this question one must be familiar with the concept of Complements. In set theory, the complements of a set A refers to elements not in A. When all sets under consideration are considered to be subsets of a given set $U,$ the absolute complement of A is the set of elements.