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Which of the following Implications are Tautologies?

1) If I drink milk, I'm healthy or If I'm healthy, I do maths.

2)If it is not the case that riches is sufficient for Mr. X to be happy then Mr. X is rich and not happy.

A) Both (1) and (2)

B) only (1)

C) only (2)

D) Neither
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1) If I drink milk I'm healthy or If I'm healthy I do maths.

        p="i drink milk"     q="i m healthy"  r="i do maths"

        (p$\rightarrow$q) $\vee$ (q$\rightarrow$ r)

        above statement will be false when both parts of $\vee$ are false

        (p$\rightarrow$q) will be False only when p=T and q=F

        so (q$\rightarrow$ r) here q=F then this will be Truth

        so whole statement is tautology

2) If it is not the case that riches is sufficient for Mr. X to be happy then Mr. X is rich and not happy.

  ~[  (Riches(x) $\rightarrow$ happy(x)) $\rightarrow$ (Riches(x) $\wedge$ ~Happy(x)) ]

    above statement will be false iff (Riches(x) $\wedge$ ~Happy(x)) is false

     (Riches(x) $\wedge$ ~Happy(x)) will be false :

           Case 1: Riches(x) = F

                        then (Riches(x) $\rightarrow$ happy(x)) will be True

                         so (Riches(x) $\rightarrow$ happy(x)) $\rightarrow$ (Riches(x) $\wedge$ ~Happy(x)) = False

                       so Negation of whole will be True

    Case 2: Happy(x)=True  $\Rightarrow$ ~Happy(x)=False $\Rightarrow$   (Riches(x) $\wedge$ ~Happy(x)) will be false

                  Happy(x) is True so  ( Riches(x) $\rightarrow$ happy(x)) will be True

                 T$\rightarrow$F = F

                  so Negation of whole will be True

so Ans: A is true both statements are Tautology

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