Propositional Logic

Question

Ans. A

Comments

Is this logic correct?

If you try to make the first statement false, you will have to make $((a\lor b)\rightarrow c)$ true and $((a\land b)\rightarrow c)$ false. Now, to make $((a\land b)\rightarrow c)$ false, we need to make $(a\land b)$ true and c false.

Let c = false, a= true and b=true.

But now, if you observe, even $((a\lor b)\rightarrow c)$ is now false. Thus, it is not possible to make $((a\lor b)\rightarrow c) \rightarrow ((a\land b)\rightarrow c)$ false. For any values of a,b and c, the implication cannot be made false.

Coming to the second case,

$((a\land b)\rightarrow c) \rightarrow ((a\lor b)\rightarrow c)$

To make this implication false, we need to make $((a\lor b)\rightarrow c)$ false. To make $((a\lor b)\rightarrow c)$ false, we need to make $a\lor b$ true and c false.

Let c=false, a=true, b=false.

Now, $((a\land b)\rightarrow c)$ turns out to be true for these values of a,b and c (as $F\rightarrow F\:is\:T$). Thus the entire implication will be false then for c=false, a=true,and b=false. So, it is not valid.

---

Only S1 is correct. For S2 we get false when a,b and c are T, F and F respectively.

Answer 1

u can consider it as follow

A	B	C
0	0	0
0	1	1
1	0	1
1	1	1

here in last row u can see logic is matching from 'AND' logic.

So, A is valid.

 

Comments

if u are understanding from provided answer then ok,otherwise comment i will try to ellaborate

---

Please elaborate a little brother

---

please elaborate

Answer 2

The logical implication is the tautology, in the tautology, we have seen only one thing (T => F) case should not arise.

we to check only T=>F condition only

try to proof T=>F on formula if u not able to prove that. it is implication

Answer 3

Given question is of type premise  implies conclusion.

If premise is true then if conclusion is true, then the statement is valid otherwise not valid.

we can use the equivalence of implies , a implies b = a'+b

using  this ,

S1: premise  becomes :(a+b)'+c =1

i.e a'b'+c=1

conclusion becomes (ab)'+c = a'+b'+c = ab'+ba'+ab+c= ab'+ba'+ 1 ( since ab+c=1)

Hence premise is true implies conclusion is true, hence S1 is valid.

 

But in S2 the premise and conclusion are reverse of those in S1, so we can derive conclusion from premise like the above ,so S2 is not valid.