college mathematics

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Given the argument and its Euler diagram below, determine whether the syllogism is valid or in valid.

All painters use brushes.
Some people who use brushes are teachers.
Some painters are teachers.
(The second sentence is suppose to have a line underneath it, and also the problem has a Venn diagram with three circles. In circle A:Use brushes, In circle B:Painters, B is located inside circle A, then circle C:Teachers, it is somewhat of a little bit inside circle A. But the question is wanting to know if the problem is valid or invalid.

  • college mathematics -

    The conclusion is invalid.

    Consider the set of people who use brushes. Some of them are painters, since all painters use brushes.

    But there is no guarantee that the brush people who are teachers include any painters.

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