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Given set A = x | 1 <= x2 < 9 and set B = 0, 1, 2, 3, 4, find the intersection of A and B, assuming x represents integers.

Mathematics

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Answer

B. {1,2}\text{B. } \{1, 2\}

Step-by-step solution

4 steps

To find the intersection of sets AA and BB, we first need to determine the elements of set AA.

  1. Determine the elements of set AA.

    Set AA is defined as {x∣1≤x2<9}\{x \mid 1 \leq x^2 < 9\}. This inequality can be broken down into two parts:

    1. 1≤x21 \leq x^2
    2. x2<9x^2 < 9

    For 1≤x21 \leq x^2:
    Taking the square root of both sides, we get 1≤x2\sqrt{1} \leq \sqrt{x^2}, which simplifies to 1≤∣x∣1 \leq |x|.
    This means x≤−1x \leq -1 or x≥1x \geq 1.

    For x2<9x^2 < 9:
    Taking the square root of both sides, we get x2<9\sqrt{x^2} < \sqrt{9}, which simplifies to ∣x∣<3|x| < 3.
    This means −3<x<3-3 < x < 3.

    Combining these conditions, we need xx such that (x≤−1x \leq -1 or x≥1x \geq 1) AND (−3<x<3-3 < x < 3).
    The values of xx that satisfy both conditions are:
    −3<x≤−1-3 < x \leq -1 or 1≤x<31 \leq x < 3.

  2. Identify the integer elements of set AA.

    Since set BB consists of integers and the options are sets of integers, we assume xx must be an integer.
    For the interval −3<x≤−1-3 < x \leq -1, the integers are {−2,−1}\{-2, -1\}.
    For the interval 1≤x<31 \leq x < 3, the integers are {1,2}\{1, 2\}.
    So, the integer elements of set AA are A={−2,−1,1,2}A = \{-2, -1, 1, 2\}.

  3. Find the intersection of set AA and set BB.

    Set A={−2,−1,1,2}A = \{-2, -1, 1, 2\}
    Set B={0,1,2,3,4}B = \{0, 1, 2, 3, 4\}
    The intersection A∩BA \cap B consists of elements that are present in both sets.
    Comparing the elements, we find that 1 and 2 are common to both sets.
    Therefore, A∩B={1,2}A \cap B = \{1, 2\}.

  4. Select the correct option.

    The calculated intersection is {1,2}\{1, 2\}, which corresponds to option B.

    The final answer is B. {1,2}\text{B. } \{1, 2\}.

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