Are sets `A= {-2,2},B={x:x in Z ,x^2 -4 =0}` equal ?Why?

Answer

To determine whether the sets \( A \) and \( B \) are equal, we will follow these steps:

### Step 1: Identify the elements of set \( A \)
Set \( A \) is given as:
\[ A = \{-2, 2\} \]
This set contains two elements: -2 and 2.

**Hint:** Remember that a set is defined by its elements, and the order does not matter.

### Step 2: Define set \( B \)
Set \( B \) is defined as:
\[ B = \{x : x \in \mathbb{Z}, x^2 - 4 = 0\} \]
This means we need to find all integer values of \( x \) that satisfy the equation \( x^2 - 4 = 0 \).

**Hint:** The notation \( \mathbb{Z} \) represents the set of all integers, which includes negative numbers, zero, and positive numbers.

### Step 3: Solve the equation \( x^2 - 4 = 0 \)
To solve for \( x \), we can rearrange the equation:
\[ x^2 = 4 \]
Taking the square root of both sides, we find:
\[ x = \pm 2 \]
This gives us two solutions: \( x = 2 \) and \( x = -2 \).

**Hint:** When solving quadratic equations, remember to consider both the positive and negative roots.

### Step 4: Write the elements of set \( B \)
From the solutions we found, we can write set \( B \) as:
\[ B = \{-2, 2\} \]
This set also contains the same two elements: -2 and 2.

**Hint:** Ensure that you list all possible values that satisfy the equation when defining the set.

### Step 5: Compare sets \( A \) and \( B \)
Now we compare the two sets:
\[ A = \{-2, 2\} \]
\[ B = \{-2, 2\} \]
Since both sets contain the exact same elements, we can conclude that:
\[ A = B \]

### Conclusion
The sets \( A \) and \( B \) are equal because they contain the same elements.

**Final Answer:** Yes, the sets \( A \) and \( B \) are equal because both sets contain the elements -2 and 2.

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