Given U = {2, 3, 4, 5, 6, 7, 8} and A = {3, 4, 8}, find: (a) A ∩ Ø = ? (b) A' ∩ U = ?
Understand the Problem
The question involves set theory and asks to find the intersection of set A with the null set (A ∩ Ø) and the intersection of the complement of set A with the universal set (A' ∩ U). The universal set U and set A are defined as follows: U = {2, 3, 4, 5, 6, 7, 8} and A = {3, 4, 8}.
Answer
(a) $\emptyset$ (b) $\{2, 5, 6, 7\}$
Answer for screen readers
(a) $A \cap \emptyset = \emptyset$ (b) $A' \cap U = {2, 5, 6, 7}$
Steps to Solve
- Find $A \cap \emptyset$
The intersection of any set with the null set is always the null set. $A \cap \emptyset = \emptyset$
- Find $A'$
The complement of A ($A'$) with respect to the universal set U is the set of all elements in U that are not in A.
$U = {2, 3, 4, 5, 6, 7, 8}$
$A = {3, 4, 8}$
$A' = {2, 5, 6, 7}$
- Find $A' \cap U$
The intersection of $A'$ and $U$ is the set of elements that are in both $A'$ and $U$. Since $A'$ is defined with respect to $U$, all elements of $A'$ are already in $U$. Thus, the intersection of $A'$ and $U$ is simply $A'$.
$A' \cap U = A' = {2, 5, 6, 7}$
(a) $A \cap \emptyset = \emptyset$ (b) $A' \cap U = {2, 5, 6, 7}$
More Information
The intersection of a set with the null set will always result in the null set because there are no common elements. The intersection of the complement of a set A with the universal set is equal to the complement of A.
Tips
A common mistake is thinking that the intersection of a set $A$ with the empty set $\emptyset$ results in the set $A$ itself, or in the universal set $U$. Remember that the intersection requires common elements, and the empty set has no elements.
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