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Let U represent the set of all people that frequently visit a certain restaurant, surveyed about which food they like best. Let B = { x ∈ U | x likes Hamburgers } C = { x ∈ U | x likes Corn Dogs } P = { x ∈ U | x likes Pizza } (Assume these sets are not disjoint.) Write the statement that refers to: P ∩ ( C ∪ B ) c .

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Answer:

P ∩ ( C ∪ B )^c={x∈U: x likes Pizza and x does not like neither Corn Dogs nor Hamburgers}

Step-by-step explanation:

To find the statement that defines the set P∩(C∪B)^c, remember the following operations between sets

  1. Union: the union of the sets X and Y is defined as X∪Y={z:z∈X or z∈Y}, it is the set of the elements to belong to X or belong to Y (it includes elements that belong to both sets)
  1. Intersection: the intersection of the sets X and Y is defined as X∩Y={z:z∈X and z∈Y}, it is the set of the elements that belong to both X and Y.
  1. Complement: if X⊆U the complement of X is the set X^c=U-X={z: z∈U and z∉X}. It is the set of all elements (in U, the "universe") that do not belong to X.

Let z∈P∩(C∪B)^c. By definition of intersection, z∈P and z∈(C∪B)^c.

Since z∈P, z likes Pizza. Now, by definition of complement, z∉C∪B. Then it is not possible that z∈C or z∈B. Thus z∉C and z∉B. Both are required, if one of these two were false, z would belong to either C or B, and thus, belong to the union (impossible). Since z∉C, z does not like Corn Dogs, and since z∉B, z does not like Hamburgers.

We conclude that z likes Pizza, and z does not like neither Corn Dogs nor Hamburgers.

The answer is written with x replacing z, but they are dummy variables so it does not matter.

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