GO Classes Weekly Quiz 5 | Propositional Logic | Question: 12

Answer 1

With 4 propositional variables, we can have a total of $2^{4}$ = 16 combinations of truth values.

To form two compound propositions (say, $\alpha$ and $\beta$) that are not logically equivalent, we need to find at least one combination of truth values of these variables for which $\alpha$ and $\beta$ must have different truth values.

Now for each combination of these variables, a compound proposition can hold 2 values → T or F (either of them but not both, by definition of a proposition)

So for 16 such combinations of truth values, the number of unique combinations we can generate for forming a well formed formula (w.f.f.) = 2x2x...2 (16 times) = $2^{16}$ = 65536 different formulae

p	q	r	s	$\alpha$	$\beta$
F	F	F	F	F	F
.				.	.
.				.	.
.				F	F
T	T	T	T	F	T

Comments

Nicely Answered.

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Answer 2

Here actually the question is asking about the no. of different truth tables possible with 4 propositional variables. Each truth table will have 2^4 rows i.e.,16 rows and each row can have 2 values(T or F) for the compound proposition. So, total no. of truth table possible is 2^16 = 65536.
Here each truth table is different from another. That means all the 65536 truth tables are not logically equivalent to each other.

Answer 3

The question is asking – “How many compound propositions are there that are not logically equivalent to each other?”

With 4 variables, we can have $2^4$ = 16 rows in the truth table.

Now, these 16 rows can be filled in 2^16 = 65536 different ways (as each row has only 2 possibilities i.e., either T or F, so using multiplication rule, total possibilities = 2x2x2x…..x2(16 times) = 2^16 = 65536)

Now, corresponding to each way of filling the truth table, we have a particular compound proposition that is different from the compound proposition obtained from the filling of truth table in a different way.

Since, there are 65536 different ways of filling the truth table, we can have 65536 different compound propositions that are NOT logically equivalent. Hence, answer is 65536.