The elements that different sets have in common are the members of the intersection of the sets. The intersection of sets LaTeX: A and LaTeX: B are denoted LaTeX: A\cap B.
The combination of all of the elements of different sets is called the union. The union of sets LaTeX: A and LaTeX: B are denoted LaTeX: A\cup B.
All of the elements in the universal set, U, which are not members of the indicated set are referred to as the complement of the set. The complement of LaTeX: A is written as LaTeX: A'.

If LaTeX: U=\left\{s,\:y,\:m,\:b,\:o,\:l,\:i,\:c\right\}, LaTeX: X=\left\{s,\:o,\:i,\:l\right\} and LaTeX: Y=\left\{c,\:o,\:l,\:i\right\} determine,
(a) LaTeX: X\cap Y
(b) LaTeX: X\cup Y
(c) LaTeX: X'
(d) LaTeX: Y'
Since the sets LaTeX: X and LaTeX: Y both contain the elements LaTeX: o,\:i and LaTeX: l, LaTeX: X\cap Y=\left\{o,\:i,\:l\right\}
Combining the elements in both sets LaTeX: X and LaTeX: Y, no repeated elements, we get that LaTeX: X\cup Y=\left\{s,\:c,\:o,\:i,\:l\right\}
All of the members of LaTeX: U that are NOT members of LaTeX: X comprise the complement of LaTeX: X. Therefore,

LaTeX: X'=\left\{y,\:m,\:b,\:c\right\}
LaTeX: Y'=\left\{s,\:y,\:m,\:b\right\}