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Find truth table p or (q and r)=(p or q) and (p or r)

Your problem `->` truth table p or (q and r)=(p or q) and (p or r)

To prove `pvv(q^^r)=(pvvq)^^(pvvr)`, we havet to first prepare the following truth table

TTT T `T=T^^T` T `T=TvvT` T `T=TvvT` T `T=TvvT` T `T=T^^T`
TTF F `F=T^^F` T `T=TvvF` T `T=TvvT` T `T=TvvF` T `T=T^^T`
TFT F `F=F^^T` T `T=TvvF` T `T=TvvF` T `T=TvvT` T `T=T^^T`
TFF F `F=F^^F` T `T=TvvF` T `T=TvvF` T `T=TvvF` T `T=T^^T`
FTT T `T=T^^T` T `T=FvvT` T `T=FvvT` T `T=FvvT` T `T=T^^T`
FTF F `F=T^^F` F `F=FvvF` T `T=FvvT` F `F=FvvF` F `F=T^^F`
FFT F `F=F^^T` F `F=FvvF` F `F=FvvF` T `T=FvvT` F `F=F^^T`
FFF F `F=F^^F` F `F=FvvF` F `F=FvvF` F `F=FvvF` F `F=F^^F`

from this table, we can say that columns (5) and (8) are identical.
`:. pvv(q^^r)=(pvvq)^^(pvvr)`

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