Sets - Disjoint Sets-advanced

Let $A = \{x : x \text{ is a prime number and } 10 < x < 20\}$ and $B = \{x : x \text{ is a multiple of } 4 \text{ and } 10 < x < 20\}$. Determine if $A$ and $B$ are disjoint sets.

First, list the elements of each set: $A = \{11, 13, 17, 19\}$ $B = \{12, 16\}$ Now find the intersection: $A \cap B = \{11, 13, 17, 19\} \cap \{12, 16\} = \emptyset$ Since $A \cap B = \emptyset$, the sets $A$ and $B$ are disjoint.

To check for disjointness, identify the individual elements of the sets based on the given conditions. If there are no common elements, their intersection is the empty set, confirming they are disjoint.

Given two disjoint sets $X$ and $Y$. If $n(X \cup Y) = 50$ and $n(X) = 22$, find the value of $n(Y)$.

For disjoint sets, we use the formula: $n(X \cup Y) = n(X) + n(Y)$ Substituting the given values: $50 = 22 + n(Y)$ $n(Y) = 50 - 22$ $n(Y) = 28$

In the case of disjoint sets, the intersection is zero. Therefore, the total number of elements in the union is simply the sum of the cardinalities of the individual sets.

If $A = \{n^2 : n \in \{1, 2, 3\}\}$ and $B = \{n^3 : n \in \{1, 2, 3\}\}$, are $A$ and $B$ disjoint? If not, modify set $B$ to make them disjoint while keeping the same number of elements.

Let's find the elements: $A = \{1^2, 2^2, 3^2\} = \{1, 4, 9\}$ $B = \{1^3, 2^3, 3^3\} = \{1, 8, 27\}$ Find the intersection: $A \cap B = \{1\} \neq \emptyset$ Thus, $A$ and $B$ are NOT disjoint. To make them disjoint while keeping $n(B) = 3$, we must remove the common element $1$. We can replace $1$ with any value not in $A$, such as $64$ (which is $4^3$). Modified $B = \{8, 27, 64\}$. Now $A \cap B = \emptyset$.

Sets are disjoint only if the intersection is empty. Since both sets shared the element $1$, they were not disjoint. Changing the overlapping element to a unique one satisfies the condition for disjoint sets.