20 votes

Which one of the following Boolean expressions is NOT a tautology?

- $((\,a\,\to\,b\,)\,\wedge\,(\,b\,\to\,c))\,\to\,(\,a\,\to\,c)$
- $(\,a\,\to\,c\,)\,\to\,(\,\sim b\,\to\,(a\,\wedge\,c))$
- $(\,a\,\wedge\,b\,\wedge\,c)\,\to\,(\,c\vee\,a)$
- $a\,\to\,(b\,\to\,a)$

31 votes

Best answer

Another way to solve it...

Implication $A\to B$ is not tautology if $B$ is false and $A$ is true.

For b option Let RHS ie. $b\to (a\wedge c)$ be false ie $b$ is false and $(a\wedge c)$ is false.

Now, $a$ AND $c$ is false if both $a$ and $c$ are false or one of them is true and other is false.

Now, if $a$ and $c$ both are false then $a\to c$ is true. Now ,LHS is $\text{true}$ and RHS is $\text{false}.$

So option b is not tautology..

Implication $A\to B$ is not tautology if $B$ is false and $A$ is true.

For b option Let RHS ie. $b\to (a\wedge c)$ be false ie $b$ is false and $(a\wedge c)$ is false.

Now, $a$ AND $c$ is false if both $a$ and $c$ are false or one of them is true and other is false.

Now, if $a$ and $c$ both are false then $a\to c$ is true. Now ,LHS is $\text{true}$ and RHS is $\text{false}.$

So option b is not tautology..

0

In option C if a and b are false and c is true then the proposition should be F --> T. Then, the proposition should be false and hence it must not be tautology. Am I correct ?

0

No,you are not correct.Conditional statement P->q is false only when p is true and q is false and true otherwise.

You can also check in this way-

(a∧b∧c)→(c∨a)

≡~(a∧b∧c)∨ (c∨a)

≡(~a∨~b∨~c)∨ (c∨a)

≡(~a∨a)∨(~b∨b)∨ c

≡T∨T∨c

≡T∨c

≡T

Hence,(a∧b∧c)→(c∨a) is tautology.

0

i am not able to get it for option a)

a) can be reduced like this:

((a`+b)(b`+c))`+a`+c

:= ab` + bc` + a` + c

now how this is a tautology?

a) can be reduced like this:

((a`+b)(b`+c))`+a`+c

:= ab` + bc` + a` + c

now how this is a tautology?

3

@ Aayushi ,

:= ab` + bc` + a` + c continuing ...

= **a'+ab'** + **c+c'b**

** **= a'+b' + c+b for details see this http://cs.stackexchange.com/questions/24587/which-law-is-this-expression-x-x-y-xy

= a'+c + b+b'

= a'+c + 1

= 1 (Tautology)

0

LeenSharma putting 0/1 can be time consuming because we may have to check all the combination and that can be frustrating in exam.

13 votes

$A.\ \ ((a \rightarrow b) \wedge (b \rightarrow c)) \rightarrow (a \rightarrow c)$

$\equiv (( \sim a \vee b) \wedge (\sim b \vee c)) \rightarrow (\sim a \vee c)$

$\equiv \sim (( \sim a \vee b) \wedge (\sim b \vee c)) \vee (\sim a \vee c)$

$\equiv (( a \ \wedge \sim b) \vee ( b \wedge \sim c)) \vee (\sim a \vee c)$

$\equiv (\sim a \vee ( a \ \wedge \sim b) )\vee( ( b \wedge \sim c) \vee c)$

$\equiv( (\sim a \vee a )\wedge(\sim a \vee \sim b) )\vee( ( b \vee c) \wedge( \sim c\vee c))$

$\equiv(T\wedge(\sim a \vee \sim b) )\vee( ( b \vee c) \wedge T)$

$\equiv\sim a \vee (\sim b \vee b) \vee c$

$\equiv\sim a \vee T \vee c$

$\equiv T$

$B.\ \ (a \rightarrow c)\rightarrow (\sim b \rightarrow(a \wedge c))$

$\equiv \sim(\sim a \vee c)\vee (( b \vee (a \wedge c))$

$\equiv ( a \wedge \sim c)\vee (( b \vee (a \wedge c))$

$\equiv (( a \wedge \sim c)\vee ( a \wedge c) )\vee b$

$\equiv (a \wedge(c \vee \sim c))\vee b$

$\equiv a \vee b $

$C. \ \ (a\wedge b \wedge c) \rightarrow(c \vee a)$

$\equiv \sim(a\wedge b \wedge c) \vee (c \vee a)$

$\equiv \sim a \sim b \sim c \vee c \vee a$

$\equiv (a\vee\sim a )\vee \sim b \vee(\sim c \vee c)$

$\equiv T\vee \sim b \vee T$

$\equiv T$

$D.\ \ a\rightarrow (b\rightarrow a)$

$\equiv\sim a \vee (\sim b \vee a)$

$\equiv(\sim a\vee a)\vee \sim b$

$\equiv T \vee \sim b$

$\equiv T$

Hence, Option(B) $(a \rightarrow c)\rightarrow (\sim b \rightarrow(a \wedge c))$ is the correct choice.

$\equiv (( \sim a \vee b) \wedge (\sim b \vee c)) \rightarrow (\sim a \vee c)$

$\equiv \sim (( \sim a \vee b) \wedge (\sim b \vee c)) \vee (\sim a \vee c)$

$\equiv (( a \ \wedge \sim b) \vee ( b \wedge \sim c)) \vee (\sim a \vee c)$

$\equiv (\sim a \vee ( a \ \wedge \sim b) )\vee( ( b \wedge \sim c) \vee c)$

$\equiv( (\sim a \vee a )\wedge(\sim a \vee \sim b) )\vee( ( b \vee c) \wedge( \sim c\vee c))$

$\equiv(T\wedge(\sim a \vee \sim b) )\vee( ( b \vee c) \wedge T)$

$\equiv\sim a \vee (\sim b \vee b) \vee c$

$\equiv\sim a \vee T \vee c$

$\equiv T$

$B.\ \ (a \rightarrow c)\rightarrow (\sim b \rightarrow(a \wedge c))$

$\equiv \sim(\sim a \vee c)\vee (( b \vee (a \wedge c))$

$\equiv ( a \wedge \sim c)\vee (( b \vee (a \wedge c))$

$\equiv (( a \wedge \sim c)\vee ( a \wedge c) )\vee b$

$\equiv (a \wedge(c \vee \sim c))\vee b$

$\equiv a \vee b $

$C. \ \ (a\wedge b \wedge c) \rightarrow(c \vee a)$

$\equiv \sim(a\wedge b \wedge c) \vee (c \vee a)$

$\equiv \sim a \sim b \sim c \vee c \vee a$

$\equiv (a\vee\sim a )\vee \sim b \vee(\sim c \vee c)$

$\equiv T\vee \sim b \vee T$

$\equiv T$

$D.\ \ a\rightarrow (b\rightarrow a)$

$\equiv\sim a \vee (\sim b \vee a)$

$\equiv(\sim a\vee a)\vee \sim b$

$\equiv T \vee \sim b$

$\equiv T$

Hence, Option(B) $(a \rightarrow c)\rightarrow (\sim b \rightarrow(a \wedge c))$ is the correct choice.