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Express each of statement using Logical operators , Predicates and Quantifiers

a) The negation of a contradiction is a Tautology

b) The disjunction of 2 contigencies can be tautology

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Let the domain be the collection of all proposition.

$P(x)=x \ is \ a \ tutology$

$Q(x)=x \ is \ a \ contradiction$

$Q(x)=x \ is \ a \ contingency$

$\sim x \ is \ the \ negation \ of \ proposition \ x.$

$(a)The \ negation \ of \ a \ contradiction \ is \ a\ tautology$

$It \ means \ " for \ all \ propositions \ x \ such \ that \ x \ is \ a \ contradiction, \\ the \ negation \ ¬x \ is \ a \ tautology".$

 

$\forall x(Q(x) \rightarrow P(\sim x))$

 

$(b)The \ disjunction \ of \ 2 \ contingencies \ can \ be\ tautology$

$It \ means \ " There \ are \ x \ and \ y \ such \ that \ x \ and \ y \ are \ contingencies \\ and\ x \vee y \ is \ a \ tautology".$
 

$\exists x \exists y((R(x) \wedge R(y))\rightarrow P(x \vee y)$

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