Express the propositions $\displaystyle p\implies q$ and $\displaystyle q\wedge p$ using only $\displaystyle \downarrow$ and $\displaystyle \sim.$

$\displaystyle p\downarrow q,$ where $\displaystyle p$ and $\displaystyle q$ are propositions, it's defined by the following truth table:

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Let $\displaystyle p_1,p_2,p_3,p_4,p_5,p_6$ be propositions such that $\displaystyle \left[ {(p_1 \veebar p_2 ) \implies (p_3 \implies p_4 )} \right]$ is false. Determine the value of truth of:

- $\displaystyle (p_5\implies p_6)\vee(p_1\vee p_2).$
- $\displaystyle [(p_5\implies p_2)\vee\sim p_1]\implies(p_4\vee p_3).$
- $\displaystyle \sim[(p_6\vee p_5)\wedge(p_1\wedge p_2)\iff(p_4\implies p_3)].$

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Let $\displaystyle p,q$ and $\displaystyle r$ be propositions such that $\displaystyle ((\sim p\vee q)\implies r)$ is false. Give the value of truth of the following propositions (justify your answer):

- $\displaystyle \sim q\implies\sim p.$
- $\displaystyle r\implies(p\iff\sim(q\vee r)).$

I'm learning these stuff, so if you can help me, it'd be great. Thanks