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Is there anyother way to solve this problem other than drawing the complete truth table ?

2 Answers

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Use Rules of Inference,

Method:1 For option1, By transitivity, $(p\rightarrow q) \wedge (q\rightarrow r) \Rightarrow (p\rightarrow r)$

Hence it is valid, apply rules and find out.

Method:2 Proving the statement results in false. That is, for Implication prove ($T\rightarrow F$) case, for conjunction $T\wedge F$ etc..

For option 2: If you prove lhs=T and rhs=F then it is invalid.

Lets start with RHS, for RHS to be false, $(-p\rightarrow -q)$, -p should be true and -q should be false, hence, P is F and Q is T.

Substitute these values in LHS:
$(F\rightarrow T\equiv T)$,

now, LHS=T, RHS=F (LHS --> RHS = T --> F which is False, hence the statement is invalid)
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You can do option reduction.. i have done for only option a.. 

For rest of the options T-->F condition will be possible.. so they are not valid.

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