Show that {~, A} is an adequate set of connectives. i.e. show that every compoun

Question

Show that {~, A} is an adequate set of connectives. i.e. show that every compound statement using primitive statements (e.g., p, q, r, etc.) and the logical connectives ~, Lambda, V, and is logically equivalent to a compound statement using the same primitive statements and only the connectives ~ and Lambda. (Note that it suffices to show that each of the compound statements p V q, p q and p q is logically equivalent to a statement using only the primitive statements p and q and the symbols ~ and Lambda, and possibly grouping parentheses.)

Explanation / Answer

p q ~p ~q (p v q) (p ^ q) (~p v q) (~p ^ ~q) (p => q) (q => p) ( p <=> q)

0 0 1 1 0 0 1 1 1 1 1

0 1 1 0 1 0 1 0 1 0 0

1 0 0 1 1 0 0 0 0 1 0   

1 1 0 0 1 1 1 0 1 1 1

From table p v q = (~p ^ ~q) --- (1)

p => q = (~p v q) = (~~p ^ ~q) = (p ^ ~q) [from table and 1] --- (2)

p <=> q = (p => q) ^ (q => p) = (p ^ ~q) ^ (q ^ ~p) [ from table and 2]

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