Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8}, N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}.Use these sets to find each of the following. Identify any disjoint sets. M ∩ N

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Recall that the intersection of two sets, denoted by $M \cap N$, consists of all elements that are common to both sets $M$ and $N$.

List the elements of set $M$: $\{0, 2, 4, 6, 8\}$ and set $N$: $\{1, 3, 5, 7, 9, 11, 13\}$.

Compare the elements of $M$ and $N$ to find any common elements. Since $M$ contains only even numbers and $N$ contains only odd numbers, check if any number appears in both sets.

If no elements are common between $M$ and $N$, then their intersection $M \cap N$ is the empty set $\emptyset$.

Since $M \cap N = \emptyset$, this means sets $M$ and $N$ are disjoint, which means they have no elements in common.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Set Intersection

The intersection of two sets consists of all elements that are common to both sets. It is denoted by the symbol ∩. For example, if A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3}.

Disjoint Sets

Two sets are disjoint if they have no elements in common, meaning their intersection is the empty set (∅). For instance, if A = {1, 2} and B = {3, 4}, then A and B are disjoint because A ∩ B = ∅.

Universal Set and Subsets

The universal set U contains all elements under consideration, while subsets like M, N, Q, and R are parts of U. Understanding the relationship between subsets and the universal set helps in analyzing intersections and disjointness.

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Perform each division. See Examples 7 and 8. $(- 4 m^{2} n^{2} - 21 m n^{3} + 18 m n^{2}) / (- 14 m^{2} n^{3})$