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\begin{document}
    \textbf{Question 1}\\\\
    - Point(x) for 'x is a point'\\
    - Line(x) for 'x is a straight line'\\
    - LiesOn(x, y) for 'x lies on y'\\
    - x=y for 'x coincides with y'\\

    1) For any two distinct points, there is a unique line containing both of them.\\
    Solution:
    $$\forall x \forall y: Point(x)\wedge Point(y)\wedge\overline{x=y}\Longrightarrow$$
    $$\Longrightarrow\exists!L: Line(L)\wedge LiesOn(x, L)\wedge LiesOn(y, L)$$

    2) The definition of parallel lines.\\
    Solution:
    $$Parallel(L_1, L_2) - \forall L_1 \forall L_2: Line(L_1)\wedge Line(L_2)\Longrightarrow$$
    $$\Longrightarrow\overline{\exists x}: Point(x)\wedge LiesOn(x, L_1)\wedge LiesOn(x, L_2)$$

    3) For any line x and any point y that is not lying on x, there exists a unique line z, that contains y and is parallel to x.\\
    Solution:
    $$\forall x: Line(x)$$
    $$\forall y: Point(y)\wedge\overline{LiesOn(y, x)}\Longrightarrow$$
    $$\Longrightarrow !\exists z: Line(z)\wedge LiesOn(y, z)\wedge Parallel(x, z)$$

    \textbf{Question 2}\\\\
    Use the technique explained in the lecture to show that the following formula is valid:
    $$\forall x(\overline{P(x)}\Rightarrow Q(x))\Rightarrow\forall y(\overline{Q(y)}\Rightarrow P(y))$$
    Solution:
    $$$$
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