Question 1190027
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The symbol ^ means "and". It refers to a logical conjunction. In logic textbooks, the symbol ^ does not mean exponent.


The symbol v means "or". It's a logical disjunction. 


Here's a truth table for p ^ q<table border = "1" cellpadding = "5"><tr><td>p</td><td>q</td><td>p ^ q</td></tr><tr><td>T</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</td><td>F</td></tr><tr><td>F</td><td>T</td><td>F</td></tr><tr><td>F</td><td>F</td><td>F</td></tr></table>T = true, F = false
As the table shows, p ^ q is only true if BOTH p and q are true together. Otherwise, it's false.


Here's a table for p v q<table border = "1" cellpadding = "5"><tr><td>p</td><td>q</td><td>p v q</td></tr><tr><td>T</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</td><td>T</td></tr><tr><td>F</td><td>T</td><td>T</td></tr><tr><td>F</td><td>F</td><td>F</td></tr></table>We have the opposite situation going on here. 
The expression p v q is only false when BOTH p and q are false together. Otherwise, it's true.


Use the ideas mentioned to get<table border = "1" cellpadding = "5"><tr><td>p</td><td>q</td><td>p ^ q</td><td>(p ^ q) v p</td></tr><tr><td>T</td><td>T</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</td><td>F</td><td>T</td></tr><tr><td>F</td><td>T</td><td>F</td><td>F</td></tr><tr><td>F</td><td>F</td><td>F</td><td>F</td></tr></table><font color=red>That table is the final answer.</font>
Notice how the final column is perfectly identical to the p column. 
Therefore, the expression (p ^ q) v p is logically equivalent to p.


If you wanted, you can have this as scratch work off to the side somewhere<table border = "1" cellpadding = "5"><tr><td>p ^ q</td><td> p</td><td>(p ^ q) v p</td></tr><tr><td>T</td><td>T</td><td>T</td></tr><tr><td>F</td><td>T</td><td>T</td></tr><tr><td>F</td><td>F</td><td>F</td></tr><tr><td>F</td><td>F</td><td>F</td></tr></table>This way you can quickly compare this scratch work table to the p v q table shown earlier. 
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