summary of classes from class 29 to 32 added

class_29_31102023/summary.txt

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+Quiz was conducted. 
+

class_30_31102023/summary.txt

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+# Proof system for FOL
+	Axiom Group1 : substitution in propositional tautologies
+	Axiom Group2 : (\forall x \alpha -> \alpha[t,x]) for any term t substituting x in valid way (that is no free variable becomes bound). 
+	Axiom Group3 : (\forall x (\alpha -> \beta) -> (\forall x \alpha -> \forall x \beta))
+	Axiom Group4 : (\alpha -> (\forall x \alpha)), when x is not a free variable in \alpha
+
+## Then Axioms=I({Axiom group1, Axiom group2, Axiom group3, Axiom group4}, {\forall x, \forall y,...})
+
+## Any formula has a proof if it belongs to I(Axioms,{MP}). 
+
+-- example of a proof. 
+
+

class_31_02112023/summary.txt

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+-- substitution
+-- soundness theorem
+-- base case proof. 
+