Question

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

• A. $((a\to b)\wedge (b\to c))\to (a\to c)$
• B. $(a\leftrightarrow c)\to (\sim b\to (a\wedge c))$
• C. $(a\wedge b\wedge c)\to (c\vee a)$
• D. $a\to (b\to a)$

Answer 1

The correct option is B $(a\leftrightarrow c)\to (\sim b\to (a\wedge c))$

$(a\leftrightarrow c)\to (\sim b\to (a\wedge c\left)\right)$
$\equiv (a\leftrightarrow c){}^{\prime }+(b{}^{\prime }\to ac)$
$\equiv (a⨁c){}^{\prime }+(b{}^{\prime }\to ac)$
$\equiv ac{}^{\prime }+a{}^{\prime }c+b+ac$
$\equiv a(c{}^{\prime }+c)+a{}^{\prime }c+b$
$\equiv a+a{}^{\prime }c+b\equiv a+c+b$
Which is not a tautology .