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Using the universal set U = 1, 2, ..., 12, let E be the set of even numbers and M be the set of multiples of 3. Find n(M), E ∩ M, and a possible value for y if y ∈ (E ∪ M).

Mathematics

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3 parts
a

n(M)=4n(M) = 4

b

E∩M={6,12}E \cap M = \{6, 12\}

c
2

Step-by-step solution

4 steps

Here are the solutions for the questions.

Part 1

Set Theory

First, let's define the sets based on the given information:
The universal set E\mathcal{E} consists of integers xx such that 1≤x≤121 \le x \le 12.

E={1,2,3,4,5,6,7,8,9,10,11,12}\mathcal{E} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}

Set EE consists of even numbers within E\mathcal{E}.

E={2,4,6,8,10,12}E = \{2, 4, 6, 8, 10, 12\}

Set MM consists of multiples of 3 within E\mathcal{E}.

M={3,6,9,12}M = \{3, 6, 9, 12\}

Part a

n(M)=4n(M) = 4

Find n(M)n(M).

  1. Count the number of elements in set MM.

    M={3,6,9,12}M = \{3, 6, 9, 12\} has 4 elements.

    The answer is n(M)=4\boxed{n(M) = 4}.

Part b

E∩M={6,12}E \cap M = \{6, 12\}

Write down the set E∩ME \cap M.

  1. Identify the elements that are common to both set EE and set MM.

    E={2,4,6,8,10,12}E = \{2, 4, 6, 8, 10, 12\}
    M={3,6,9,12}M = \{3, 6, 9, 12\}
    The common elements are 6 and 12.

    The answer is E∩M={6,12}\boxed{E \cap M = \{6, 12\}}.

Part c2

y∈(E∪M)y \in (E \cup M). Write down a possible value of yy.

  1. Find the union of set EE and set MM, which includes all elements in EE or MM (or both).

    E∪M={2,3,4,6,8,9,10,12}E \cup M = \{2, 3, 4, 6, 8, 9, 10, 12\}

  2. Choose any element from the set E∪ME \cup M.

    For example, 2.

    A possible value of yy is 2\boxed{2}. (Other valid answers include 3,4,6,8,9,10,123, 4, 6, 8, 9, 10, 12).

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