Recall that two statements are logically equivalent if they contain the same col
ID: 3110790 • Letter: R
Question
Recall that two statements are logically equivalent if they contain the same column configuration on a truth table. If two statements are equivalent we say they belong to the same equivalence class. There are an infinite number of statements which can be constructed using the variable p and logical connectives. For example: p rightarrow p, p p, (p ~p) rightarrow (p p), etc. However, it can be shown that all possible statements constructed with just one variable letter are all logically equivalent to one of four statements; i.e. there are only four equivalence classes for statements formed by one variable letter. For example: all statements formed from the single variable p and logical connectives are equivalent to one of the following statements: p, ~p, the contradiction p ~p, and the tautology p ~p. How many equivalence classes are there for statements formed with 5 letters, i.e. how many non-logically equivalent statements consist of 5 variable letters?Explanation / Answer
According to given information, there are only 4 (2^1 * 2) equivalence classes for statements formed by one variable letter i.e. p, ~p , p^~p and pV~p.
So lets suppose we take 2 letters p and q
The no of statements formed by these two letters can be infinite , for example :- (pVq) ->(p^~q) , (p^q)->(~qVp),etc
But the equivalence classes formed by these two are 2^2+2^2=2^2 * 2 =8 equivalence classes :-
1. p^q
2. p^~q
3. ~p^q
4. ~p^~q
5. pVq
6. pV~q
7. ~pVq
8. ~pV~q
Similarly we can say for 5 letters no of equivalence classes formed are 2^5 * 2 =64
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