\(\mathcal{M} = \langle D, I\rangle\) with domain:

\[\begin{align*} D = \{a,b,c,d,e\} \end{align*}\]

Interpretation function:

\[\begin{align*} I(\con{anna}) &= a\\ I(\con{ben}) &= b\\ I(\con{cara}) &= c\\ I(\con{student}) &= \{a,b\}\\ I(\con{linguist}) &= \{b,c\}\\ I(\con{book}) &= \{d,e\}\\ I(\con{admire}) &= \{(a,b),(b,c),(c,d)\}\\ I(\con{read}) &= \{(a,d),(b,e),(c,d),(a,e)\}\\ I(\con{recommend}) &= \{(a,d),(c,a),(c,d)\}\\ \end{align*}\]

Environment:

\[\begin{align*} g=\lbrace (x,a), (y,b)\rbrace \end{align*}\]

  1. State whether each atomic formula is true or false in :

    \[\begin{gather*} \con{student}\con{anna}\quad\checkmark \\ \con{book}\con{anna}\quad\times\\ \con{linguist}\con{cara}\quad\checkmark\\ \con{admire}\con{ben}\con{anna}\quad\checkmark\\ \con{admire}\con{anna}\con{ben}\quad\times\\ \end{gather*}\]

  2. Evaluate the following formulas in .

    \[\begin{gather*} \interp{\exists x.\con{student}x} = 1\\ \interp{\forall x.\con{student}x \rightarrow \con{linguist}x} = 0\\ \interp{\exists x.\con{student}x \land \con{linguist}x} = 1\\ \interp{\forall x.\con{book}x \rightarrow \con{student}x} = 0\\ \interp{\exists x\forall y. \con{admire}y\cnct x} = 0\\ \interp{\con{student}x} = 1\\ \interp{\con{linguist}x} = 0\\ \interp{\exists x.\con{admire}x\cnct y} = 1 \\ \end{gather*}\]
  3. Translate the following sentences into FOL:
    1. Anna is a student.
    \[student'a'\]
    1. Ben read a book.
    \[\exists x. book'x \land read'x\cnct{b'}\]
    1. Every student read a book.
    \[\forall x. student'x \to \exists y. book'y \land read'y\cnct x\]
    1. Some linguist recommended every book.
    \[\exists x. linguist'x \land \forall y. book'y \to recom'y\cnct x\]

    or, possibly,

    \[\forall x. book'x \to \exists y.linguist'y \land recom'x\cnct y\]
    1. No student recommended Ben.
    \[\neg\exists x. student'x \land recom'b'x\]
    1. Not every student read a book.
    \[\exists x. student'x \land \neg\exists y. book'y \land read'y\cnct x\]
    1. No book is read by every student.
    \[\neg \exists x. book'x \land \forall y. student'y \to read'x\cnct y\]
    1. Some book is read by every student.
    \[\exists x. book'x \land \forall y. student'y \to read'x\cnct y\]
  4. Translate the following sentences into FOL; you are not allowed to use a cardinality predicate:
    1. Exactly one student laughed.
    \[\exists x. student'x \land laugh'x \land \forall y. student'y \land laugh'y \to y = x\]
    1. At least two students laughed.
    \[\begin{align} \exists x,y. x\neq y\land student'x \land laugh'x \land student'y \land laugh'y\land \\ \nonumber \forall z. student'z \land laugh'z \to z = x \lor z = y \end{align}\]
    1. No more than one student laughed.
    \[\forall x,y. student'x \land laugh'x \land student'y \land laugh'y \to x = y\]