Mathematical logic and formal semantics questions.

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#	Question
1	Write a rcursive definition of the subformula $Sub(varphi )$ a proporsition $varphi$ . Mathematical Calculation
2	Determine $Sub(varphi )$ of the proporsition $varphi =(^-p_2 ightarrow (p_3vee(p_1leftrightarrow p_2)))wedge ^{-}p_3$ . Mathematical Calculation
3	Without using truth table method, prove the identity $pwedge (qvee r)=(pwedge q)vee (pwedge r)$ . Mathematical Calculation
4	Determine the parsing tree of the proporsition $(^-p_2 ightarrow (p_3vee (p_1leftrightarrow p_2)))wedge^-p_3.$ Mathematical Calculation
5	State the duality property in proporsional logic Mathematical Calculation
6	Write the dual of the identity $(pvee q)wedge (^-pvee r)wedge (qvee r)=(pvee q)wedge(^-pvee r)$ . Mathematical Calculation
7	Define the following terms as used in the studies of mathematical logic Satisfiability Validity Contradictory Mathematical Calculation
8	Define ,in word , the following natural deduction rules .For each rule , present it symbolically. Conjuction Modus Tollens Implies-elimination Implies-introduction Mathematical Calculation
9	Use the following rules to prove the given expressions below Conjuction Modus Tollens Implies-elimination Implies-introduction $s ightarrow (rvee p), (p ightarrow (^-r ightarrow q))vdash s ightarrow(qvee r)$ $((pvee q)vee r)vdash pvee(qvee r)$ Mathematical Calculation
10	Let $Q(x,y)$ be the statement $"x+y=x-y"$ . If the universe of discourse for both variables is the set of integers, what are the truth values of: $Q(1,1)$ $Q(2,0)$ $exists xQ(x,2)$ $exists yQ(3,y)$ $exists xforall yQ(x,y)$ $exists yforall xQ(x,y)$ Mathematical Calculation