Which of the following pairs of sets are equal? Justify your answer. (i) X, the set of letters in “ALLOY” and B, the set of letters in “LOYAL”. (ii) A = \{n : n $\in Z\text{ and } n^2 = 4$} and B = \{x : x $\in R\text{ and } x^2 - 3x + 2 = 0$}.

Step 1: Determine the elements of set X and set B for part (i)

To determine if two sets are equal, we first need to list all the distinct elements in each set. For part (i), set X consists of the unique letters in the word "ALLOY", and set B consists of the unique letters in the word "LOYAL". We extract each unique letter to form the sets.

Step 2: Compare set X and set B for part (i)

Two sets are considered equal if they contain exactly the same elements, regardless of the order in which the elements are listed. Since both set X and set B contain the same four letters (A, L, O, Y), they are equal.

Step 3: Determine the elements of set A for part (ii)

For part (ii), set A is defined as all integers $n$ such that $n^2 = 4$. We solve this equation to find the possible integer values for $n$. The solutions are $n = 2$ and $n = -2$. Both of these are integers, so they are included in set A.

Step 4: Determine the elements of set B for part (ii)

Set B is defined as all real numbers $x$ such that $x^2 - 3x + 2 = 0$. We solve this quadratic equation by factoring. The factors are $(x-1)$ and $(x-2)$, which gives us the solutions $x=1$ and $x=2$. Both of these are real numbers, so they are included in set B.

Step 5: Compare set A and set B for part (ii)

Now we compare the elements of set A and set B. Set A contains $\{-2, 2\}$ and set B contains $\{1, 2\}$. Since set A contains $-2$ but not $1$, and set B contains $1$ but not $-2$, the two sets do not contain exactly the same elements. Therefore, set A and set B are not equal.

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