Solution: Translation

  1. No one speaks French. (Note: you can take speak French as a single lexical item. But, if you like, try to take the two words separately.)

    \(\neg\exists x.sf'x\)

    Decomposing speak French:

    \(\neg\exists y. speak'french'y\)

  2. No one speaks French or Arabic:

    \(\neg\exists x. sf'x \lor sa'x\) \

    No one speaks French and Arabic:

    \(\neg\exists x. sf'x \land sa'x\)

  3. Every student is happy.

    \(\forall x.student'x \to happy'x\)

  4. Every student is happy and tired.

    \(\forall x.student'x \to happy'x \land tired'x\)

  5. Every student is happy, and every student is tired.

    \((\forall x.student'x \to happy'x)\land (\forall x.student'x \to tired'x)\)

  6. Every student is happy or tired.

    \(\forall x.student'x \to happy'x \lor tired'x\)

  7. Every student is happy, or every student is tired.

    \((\forall x.student'x \to happy'(x))\lor (\forall x.student'x \to tired'x)\)