Given U={f,g,h,p,q,r} and C={h,q,r}, find: (a) C ∩ ∅ (b) C' ∪ U
Understand the Problem
The question defines a universal set U and a set C, both containing elements. It then asks to find the intersection of set C with the empty set, and the union of the complement of C with the universal set U. We must determine the elements belonging to each resulting set and write the answer in roster form.
Answer
(a) $\emptyset$ (b) $\{f, g, h, p, q, r\}$
Answer for screen readers
(a) $C \cap \emptyset = \emptyset$ (b) $C' \cup U = {f, g, h, p, q, r}$
Steps to Solve
- Find $C \cap \emptyset$
The intersection of any set with the empty set is always the empty set. This is because the intersection contains elements that are in both sets, and the empty set has no elements.
Therefore, $C \cap \emptyset = \emptyset$
- Find $C'$
The complement of set $C$, denoted by $C'$, consists of all elements in the universal set $U$ that are not in $C$.
$U = {f, g, h, p, q, r}$ $C = {h, q, r}$
Thus, $C' = {f, g, p}$
- Find $C' \cup U$
The union of $C'$ and $U$, denoted by $C' \cup U$, consists of all elements that are in either $C'$ or $U$ or in both. Since $C'$ is a subset of $U$, the union of $C'$ and $U$ is simply $U$.
$C' = {f, g, p}$ $U = {f, g, h, p, q, r}$
Therefore, $C' \cup U = {f, g, h, p, q, r}$
(a) $C \cap \emptyset = \emptyset$ (b) $C' \cup U = {f, g, h, p, q, r}$
More Information
The intersection of a set with the null set is always the null set. The union of the complement of a set with the universal set is always the universal set.
Tips
A common mistake is to think that the intersection of a set and the empty set is the original set, rather than the empty set. Also, people can get tripped up performing set operations if they do not understand set theory.
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