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O Let p, q, rand that propositions:and s be four propositions. Assuming that p and r are false and s are true, find the truth value of each of the following \text { i. }((p \wedge \neg q) \rightarrow(q \wedge r)) \rightarrow(s \vee \neg q) \text { ii. }((p \vee q) \wedge(q \vee s)) \rightarrow((\neg r \vee p) \wedge(q \vee s)) ) Let p, q and be three propositions defined as follows: p means Sofia is happy, q means 'Sofia paints a picture whereas means ' Samir is happy Express each of the three following compound propositions symbolically by using p, q and r, and appropriate logical symbols. i. 'If Sofia is happy and paints a picture then Samir isn't happy' ii. 'Sofia is happy only if she paints a picture' iii. 'Sofia either paints a picture or she is not happy' Give the contrapositive, the converse and the inverse of each of the following statement: \text { i. } \forall x \in \mathbb{R} \text {, if } x>3 \text { then } x^{2}>9 \text { ii. } \forall x \subset \mathbb{R}_{\text {, if }} x(x, 1)>0 \text { then } x>0 \text { or } x<1 A tautology is a proposition that is always true. Let p. g and be three((p^q) →r) is a tautology.propositions,show that (p→ (q Vr)

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