Let $$ M=\{10,12,14,16,18\}, Q=\{10,12,14,16,18,20,22\} $$, and $$ R=\{10,11,12,13,14\} $$. Find ( $$ Q \cap M) \cup R $$. Identify any disjoint sets.

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UpStudy AI · Accepted Answer

Let's break down the problem step by step.

### Given Sets:
- $$ M = \{10, 12, 14, 16, 18\} $$
- $$ Q = \{10, 12, 14, 16, 18, 20, 22\} $$
- $$ R = \{10, 11, 12, 13, 14\} $$

### Step 1: Find $$ Q \cap M $$
The intersection $$ Q \cap M $$ consists of elements that are common to both $$ Q $$ and $$ M $$.

$$
Q \cap M = \{10, 12, 14, 16, 18\}
$$

### Step 2: Compute $$ (Q \cap M) \cup R $$
Now, take the union of $$ Q \cap M $$ with $$ R $$:

$$
(Q \cap M) \cup R = \{10, 12, 14, 16, 18\} \cup \{10, 11, 12, 13, 14\}
$$

Combining all unique elements:

$$
(Q \cap M) \cup R = \{10, 11, 12, 13, 14, 16, 18\}
$$

### Step 3: Identify Disjoint Sets
Two sets are **disjoint** if they have no elements in common. Let's check the intersections:

- **$$ M $$ and $$ Q $$:**
  $$
  M \cap Q = \{10, 12, 14, 16, 18\} \neq \emptyset
  $$
  
- **$$ M $$ and $$ R $$:**
  $$
  M \cap R = \{10, 12, 14\} \neq \emptyset
  $$
  
- **$$ Q $$ and $$ R $$:**
  $$
  Q \cap R = \{10, 12, 14\} \neq \emptyset
  $$

Since all pairwise intersections are non-empty, **there are no disjoint sets** among $$ M $$, $$ Q $$, and $$ R $$.

### **Final Answer:**
$$
(Q \cap M) \cup R = \{10,\, 11,\, 12,\, 13,\, 14,\, 16,\, 18\}
$$
Additionally, no two of the sets $$ M $$, $$ Q $$, and $$ R $$ are disjoint.
$$ (Q \cap M) \cup R = \{10, 11, 12, 13, 14, 16, 18\} $$ There are no disjoint sets among $$ M $$, $$ Q $$, and $$ R $$.

Let , and . Find ( . Identify any disjoint sets.

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