Question 1102118
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Notes about notation:
The tilde indicates "not". Example: ~A = not A
The arrow "->" without quotes is the implication "if..then..". For example writing "A -> B" means "if A, then B"
The ampersand & represents conjunction. Example: A & B = A and B
The vertical bar | represents disjunction "or". Example: A | B = A or B. I would use a lowercase 'v' for the disjunction operator, but I'm using a vertical bar to avoid confusion with the V already there



Here is one way to do the derivation:
<table border=1><tr><th>Number</th><th>Statement</th><th>Lines Used</th><th>Reason</th></tr><tr><td>1</td><td>Q -&gt; (A &amp; V)</td><td></td><td></td></tr><tr><td>2</td><td>(~V -&gt; C) -&gt; B</td><td></td><td></td></tr><tr><td>:.</td><td>Q -&gt; B</td><td></td><td></td></tr><tr><td>3</td><td>~Q | (A &amp; V)</td><td>1</td><td>Material Implication</td></tr><tr><td>4</td><td>(~Q | A) &amp; (~Q | V)</td><td>3</td><td>Distribution</td></tr><tr><td>5</td><td>(Q -&gt; A) &amp; (Q -&gt; V)</td><td>4</td><td>Material Implication</td></tr><tr><td>6</td><td>(Q -&gt; V) &amp; (Q -&gt; A)</td><td>5</td><td>Commutation</td></tr><tr><td>7</td><td>Q -&gt; V</td><td>6</td><td>Simplification</td></tr><tr><td>8</td><td>~(~V -&gt; C) | B</td><td>2</td><td>Material Implication</td></tr><tr><td>9</td><td>~(V | C) | B</td><td>8</td><td>Material Implication</td></tr><tr><td>10</td><td>(~V &amp; ~C) | B</td><td>9</td><td>De Morgan's Law</td></tr><tr><td>11</td><td>B | (~V &amp; ~C)</td><td>10</td><td>Commutation</td></tr><tr><td>12</td><td>(B | ~V) &amp; (B | ~C)</td><td>11</td><td>Distribution</td></tr><tr><td>13</td><td>B | ~V</td><td>12</td><td>Simplification</td></tr><tr><td>14</td><td>~V | B</td><td>13</td><td>Commutation</td></tr><tr><td>15</td><td>V -&gt; B</td><td>14</td><td>Material Implication</td></tr><tr><td>16</td><td>Q -&gt; B</td><td>7,15</td><td>Hypothetical Syllogism</td></tr></table>

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