Question 1030232
<table border=1 cellpadding=3><tr><th colspan="2">Number</th><th>Statement</th><th>Lines Used</th><th>Reason</th><th>Notes</th></tr><tr><td>1</td><td></td><td>(A v B) &gt; (C &amp; D)</td><td></td><td></td><td></td></tr><tr><td>2</td><td></td><td>(C v E) &gt; ~B</td><td></td><td></td><td></td></tr><tr><td>3</td><td></td><td>(D v F) &gt; ~A</td><td></td><td></td><td></td></tr><tr><td>:.</td><td></td><td>~A &amp; ~B</td><td></td><td></td><td></td></tr><tr><td></td><td>4</td><td>~(~A &amp; ~B)</td><td></td><td>AIP</td><td></td></tr><tr><td></td><td>5</td><td>~~A v ~~B</td><td>4</td><td>DM</td><td></td></tr><tr><td></td><td>6</td><td>A v B</td><td>5</td><td>DN</td><td></td></tr><tr><td></td><td>7</td><td>C &amp; D</td><td>1,6</td><td>MP</td><td></td></tr><tr><td></td><td>8</td><td>C</td><td>7</td><td>Simp</td><td></td></tr><tr><td></td><td>9</td><td>D</td><td>7</td><td>Simp</td><td></td></tr><tr><td></td><td>10</td><td>C v E</td><td>8</td><td>Add</td><td></td></tr><tr><td></td><td>11</td><td>~B</td><td>2,10</td><td>MP</td><td></td></tr><tr><td></td><td>12</td><td>D v F</td><td>9</td><td>Add</td><td></td></tr><tr><td></td><td>13</td><td>~A</td><td>3,12</td><td>MP</td><td></td></tr><tr><td></td><td>14</td><td>~A &amp; ~B</td><td>13,11</td><td>Conj</td><td></td></tr><tr><td></td><td>15</td><td>[~(~A &amp; ~B)] &amp; [~A &amp; ~B]</td><td>4,14</td><td>Conj</td><td></td></tr><tr><td>16</td><td></td><td>~A &amp; ~B</td><td>4-15</td><td>IP</td><td>See note below</td></tr></table>


Note: Line 4 and line 14 contradict one another. So IF you make the assumption <font color=blue>~(~A & ~B)</font> (as done on line 4) then it leads to a contradiction (on line 14). That invalidates the iniatial assumption making the opposite of the assumption true. The opposite of <font color=blue>~(~A & ~B)</font> is <font color=blue>~~(~A & ~B)</font> which turns into <font color=blue>~A & ~B</font>



Abbreivations Used:


Add: Addition
AIP: Assumption for Indirect Proof
Conj: Conjunction
DM: De Morgan's Law
DN: Double Negation
IP: Indirect Proof (aka proof by contradiction)
MP: Modus Ponens
Simp: Simplification