{"id":12205,"date":"2021-11-09T12:21:29","date_gmt":"2021-11-09T12:21:29","guid":{"rendered":"http:\/\/www.kaptest.com\/blog\/prep\/?p=12205"},"modified":"2021-11-09T19:17:09","modified_gmt":"2021-11-09T19:17:09","slug":"formulas-for-set-theory","status":"publish","type":"post","link":"https:\/\/wpapp.kaptest.com\/study\/gmat\/formulas-for-set-theory\/","title":{"rendered":"GMAT Quantitative: Formulas for Set Theory"},"content":{"rendered":"

Some tougher GMAT<\/a> Quantitative questions<\/a> will require you to know the formulas for set theory, presenting two or three sets and asking various questions about them. To refresh, the union of sets is all elements from all sets. The intersection of sets is only those elements common to all sets.<\/p>\n

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Formula for Two Overlapping Sets<\/h2>
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\nA classic GMAT setup involves a large group that is subdivided into two potentially overlapping subgroups. For example, let’s say that in a room of 20 people, there are 12 dog owners and 14 cat owners. Since 12 plus 14 is more than 20, the only way this situation makes any sense is if some people own both a dog and a cat. And it’s possible that some own neither.<\/p>\n

Essentially, there are four different subgroups to consider:<\/p>\n

    \n
  1. Those who own a dog but not a cat<\/li>\n
  2. Those who own a cat but not a dog<\/li>\n
  3. Those who own both a cat and a dog<\/li>\n
  4. Those who own neither a cat nor a dog<\/li>\n<\/ol>\n

    You could also combine some of these groups to consider both the total number of dog owners and the total number of cat owners.<\/p>\n

    You can often use the overlapping set formula to solve questions related to these kinds of setups:<\/p>\n

    Group 1 + Group 2 – Both + Neither = Total<\/p>\n

    Overlapping Set Forumla Example<\/h3>
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    An office manager orders 27 pizzas for a party. Of these, 15 have pepperoni, and 10 have mushrooms. If 4 pizzas have no toppings at all, and no other toppings are ordered, then how many pizzas were ordered with both pepperoni and mushrooms?<\/p>\n

    Group 1 + Group 2 – Both + Neither = Total<\/p>\n

    Pepperoni + Mushroom – Both + Neither = Total<\/p>\n

    15 + 10 – Both + 4 = 27<\/p>\n

    29 – Both = 27<\/p>\n

    Both = 2
    \n

    Formulas for Three Sets<\/h2>
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    \nLet\u2019s call our sets A, B, and C.<\/p>\n

    If\u00a0n = intersection<\/em>\u00a0and\u00a0u = union<\/em>.<\/p>\n

    Here are the need-to-know formulas:<\/p>\n

    P(A u B u C) = P(A) + P(B) + P(C) \u2013 P(A n B) \u2013 P(A n C) \u2013 P(B n C) + P(A n B n C)<\/p>\n

    To find the number of people in\u00a0exactly<\/em>\u00a0one set:<\/p>\n

    P(A) + P(B) + P(C) \u2013 2P(A n B) \u2013 2P(A n C) \u2013 2P(B n C) + 3P(A n B n C)<\/p>\n

    To find the number of people in\u00a0exactly<\/em>\u00a0two sets:<\/p>\n

    P(A n B) + P(A n C) + P(B n C) \u2013 3P(A n B n C)<\/p>\n

    To find the number of people in\u00a0exactly<\/em> three sets:<\/p>\n

    P(A n B n C)<\/p>\n

    To find the number of people in\u00a0two or more<\/em>\u00a0sets:<\/p>\n

    P(A n B) + P(A n C) + P(B n C) \u2013 2P(A n B n C)<\/p>\n

    To find the number of people in\u00a0at least<\/em>\u00a0one set:
    \n\"set<\/a>
    \nP(A) + P(B) + P(C) – P(A n B) – P(A n C) – P(B n C) + 2 P(A n B n C)<\/p>\n

    For questions involving set theory, it may be helpful to make a Venn diagram to visualize the solution.<\/p>\n

    To find the union of all set: (A + B + C + X + Y + Z + O)<\/p>\n

    Number of people in\u00a0exactly<\/em> one set: (A + B + C)<\/p>\n

    Number of people in\u00a0exactly<\/em>\u00a0two of the sets: (X + Y + Z)<\/p>\n

    Number of people in\u00a0exactly<\/em>\u00a0three of the sets: O<\/p>\n

    Number of people in\u00a0two or more<\/em>\u00a0sets: (X + Y + Z\u00a0 + O)<\/p>\n

    [ KEEP STUDYING:\u00a0Simple Quantitative Strategies for the GMAT<\/a>\u00a0]<\/strong><\/p>\n

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