Are the sets `A={4,5,6}` and `B={x:x^(2) - 5x -6=0}` disjoint?

Answer

To determine if the sets \( A = \{4, 5, 6\} \) and \( B = \{x : x^2 - 5x - 6 = 0\} \) are disjoint, we need to follow these steps:

### Step 1: Solve the equation for set B
We start by solving the quadratic equation \( x^2 - 5x - 6 = 0 \).

To factor the equation, we look for two numbers that multiply to \(-6\) (the constant term) and add up to \(-5\) (the coefficient of \( x \)). The numbers \(-6\) and \(1\) meet these criteria.

Thus, we can rewrite the equation as:
\[ (x - 6)(x + 1) = 0 \]

### Step 2: Find the roots of the equation
Setting each factor equal to zero gives us the solutions:
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

So, the set \( B \) contains the elements:
\[ B = \{-1, 6\} \]

### Step 3: Identify the elements of sets A and B
Now we have:
\[ A = \{4, 5, 6\} \]
\[ B = \{-1, 6\} \]

### Step 4: Determine the intersection of sets A and B
The intersection of two sets \( A \) and \( B \) is the set of elements that are common to both sets. We can find the intersection \( A \cap B \):
- The common elements between \( A \) and \( B \) are checked:
- \( 4 \) is not in \( B \)
- \( 5 \) is not in \( B \)
- \( 6 \) is in both \( A \) and \( B \)

Thus, the intersection is:
\[ A \cap B = \{6\} \]

### Step 5: Conclusion about disjoint sets
Since the intersection \( A \cap B \) is not empty (it contains the element \( 6 \)), we conclude that the sets \( A \) and \( B \) are not disjoint.

### Final Answer
The sets \( A \) and \( B \) are not disjoint.

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