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 ∈Z and n2\eq4andB={x:x ∈R and x2$ - 3x + 2 = 0\}.
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Step-by-Step Solution
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 n2=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 x2−3x+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.