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Mathematical Sets on the PSAT/NMSQT

Do you collect stamps, bottle caps, or tissues sneezed in by famous people? If so, your collection is a set. The PSAT/NMSQT doesn’t care about the stuff you display on your wall (though a mental health professional may want to know more about your interest in tissues). The exam evaluates how well you deal with mathematical sets. No worries — all you need to remember are a few facts:

  • The elements of a set are enclosed by brackets:{–2, –1, 0, 1, 4, 6, 7}

  • If the set continues, you see three dots after the last element: {2, 4, 6, 8 . . .}

  • A set with nothing in it — not even one element — is called an empty set and may be represented by brackets with nothing between them: { }. An empty set is usually represented by a crossed-out zero:

  • To find the union of two sets, put them together and then cross out any elements that show up more than once. For example, the union of {5, 5.5, 6, 6.5} and {6, 7, 8} is {5, 5.5, 6, 6.5, 7, 8}.

  • To find the intersection of two sets, see which elements they have in common. In the preceding bullet, the intersection of the two sets is {6}, because that’s the only common element. If two sets have no common elements, the intersection is an empty set.

If all the PSAT/NMSQT asked you to do was to look at lists of numbers, set questions would be no-brainers. However, they favor questions like “what is the intersection of the set of two-digit prime numbers less than 19 and the set of odd numbers from 11 to 35?” The answer, by the way, is {11, 13, 17}.

Try these two set questions:

  1. How many elements are the intersection of the sets {E, G, I, R} and {I, K, R, S, T}?

        (A)    1
        (B)    2
        (C)    4
        (D)    5
        (E)    7
  2. Set A = {1, 2, 3, 4, 5}, set B = {2, 4, 6, 8}, and set C = {2, 3, 5, 7}. Which element is not in the union of sets A and C?

        (A)    2
        (B)    3
        (C)    4
        (D)    5
        (E)    6

Now check your answers:

  1. B. 2

    The intersection of two sets is the elements in common. Both sets have the letters I and R, so these two sets have 2 elements in the intersection, Choice (B).

  2. E. 6

    The easiest way to do this problem is to simply figure out what the union of sets A and C is, and then look for the answer choice that doesn’t fit. A is {1, 2, 3, 4, 5}, and set C adds only the number 7 into the mix (recall that the union of two sets is the inclusive one — the operation that includes all the numbers in either set).

    So, the union of A and C is {1, 2, 3, 4, 5, 7}. Among the answer choices, Choice (E) is the only one that doesn’t belong.

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