For each pair, determine the logical relation between the left and right formulas (implies, contradicts, or independent):

\[\begin{gather} \forall x.\con{p}x\rightarrow \con{q}x& \exists x.\con{p}x\\ \forall x.\con{p}x\rightarrow \con{q}x& \exists x.\con{q}x\\ \exists x.\con{p}x\wedge \con{q}x& \exists x.\con{p}x\\ \forall x.\con{p}x &\exists x.\con{p}x\\ \neg\exists x\,\con{r}x& \forall x.\neg\con{r}x\\ \forall x.\con{p}x\land\con{r}x& \forall x.\con{p}x\rightarrow \con{r}x\\ \forall x.\con{p}x\land\con{r}x& \forall x.\con{p}x :\land. \forall x.\con{r}x\\ \forall x.\con{p}x\lor\con{r}x& \forall x.\con{p}x :\lor. \forall x.\con{r}x\\ \exists x.\con{p}x\land\con{r}x& \exists x.\con{p}x :\land. \exists x.\con{r}x\\ \exists x.\con{p}x\lor\con{r}x& \exists x.\con{p}x :\lor. \exists x.\con{r}x\\ \end{gather}\]

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

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

  2. Evaluate the following formulas in .

    \[\begin{gather*} \exists x.\con{student}x\\ \forall x.\con{student}x \rightarrow \con{linguist}x\\ \exists x.\con{student}x \land \con{linguist}x\\ \forall x.\con{book}x \rightarrow \con{student}x\\ \exists x\forall y. \con{admire}y\cnct x\\ \con{student}x\\ \con{linguist}x\\ \exists x.\con{admire}x\cnct y\\ \end{gather*}\]
  3. Translate the following sentences into FOL:
    1. Anna is a student.
    2. Ben read a book.
    3. Every student read a book.
    4. Some linguist recommended every book.
    5. No student recommended Ben.
    6. Not every student read a book.
    7. No book is read by every student.
    8. Some book is read by every student.
  4. Translate the following sentences into FOL; you are not allowed to use a cardinality predicate:
    1. Exactly one student laughed.
    2. At least two students laughed.
    3. No more than one student laughed.
  1. Build a model with exactly three individuals where the following are all true:
    • \[\exists x\,\con{student}x\]
    • \[\exists x\,\con{linguist}x\]
    • \[\neg\forall x\,\con{student}x\]
    • \[\exists x(\con{student}x\wedge \con{linguist}x)\]
    • \[\exists x\exists y\, \con{admire}(x,y)\]
  2. Build a model where the following set is inconsistent, and explain why:
    • \[\forall x(\con{student}x\rightarrow \con{read}(x,b))\]
    • \[\exists x\,\con{student}x\]
    • \[\forall x\,\neg \con{read}(x,b)\]
  3. Build a model where these are all true:
    • Every student admires some linguist.
    • No linguist admires Anna.
    • Anna admires Ben.