**Express the negations of these propositions using quantifiers, and in English. a) Every student in this class likes mathematics. b) There is a student in this class who has never seen a computer. c) There is a student in this class who has taken every mathematics course offered at this school. d) There is a student in this class who has been in at least one room of every building on campus**

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Solution:

a. $\forall x$ ( Likes (x, mathematics ) . Its negation:$\exists$ x($\neg$ Likes(x, mathematics )). i.e., Some student in this class do not like mathematics.

b. $\exists$ x( Notseen (x, computer )) . Negation: $\forall x$ $\neg$( Notseen (x, computer )). i.e., Every student in this class has seen a computer.

c. $\exists$ x $\forall y$(taken(x, y)). Negation: $\forall x$ $\exists$ y $\neg$(taken(x, y)). i.e., Every student in this class has not taken a mathematics course offered at this school.

d. $\exists$ x $\forall y$ $\exists$ z( within (z, y) $\wedge$ attended (x, z)). Negation: $\forall x$ $\exists$ y $\forall z$ $\neg$( within (z, y) $\wedge$ attended (x, z)). i.e., Every students in this class has not been in all the rooms of a building on campus.