document.write( "Question 475363: Construct a truth table for (p Λ ~ q) ↔ q \n" ); document.write( "

Algebra.Com's Answer #325996 by Theo(13342) $\"\"$ $\"About$

You can put this solution on YOUR website!
here's your truth table:
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document.write( "         p       ~q        (p^~q)    q     (p^~q)<->q\r\n" );
document.write( "         T        T           T      F          F\r\n" );
document.write( "         T        F           F      T          F\r\n" );
document.write( "         F        T           F      F          T\r\n" );
document.write( "         F        F           F      T          F\r\n" );
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\n" ); document.write( "Your AND relationship (p^~q) requires both p and ~q to be true in order for it to be true.
\n" ); document.write( "That's why you only see one T in the (p^~q) column (third column).
\n" ); document.write( "The IF AND ONLY IF relationship, otherwise known as the EQUIVALENCE relationship, requires both (p^~q) and q to be True or both (p^~q) and q to be False in order for it to be true. Any other combination results in False.
\n" ); document.write( "Since (p^~q) and q can never both be true at the same time, then this relationship will only be true if they're both false.
\n" ); document.write( "You can see that (p^~q)<->q is true when both (p^~q) and q are both false.
\n" ); document.write( "The rules of the IMPLIES statement (a->b) and the EQUIVALENCE statement (a<->b) statements are difficult to understand.
\n" ); document.write( "Why a->b is true when a is false regardless of whether b is true or false is a difficult concept to grasp.
\n" ); document.write( "Similarly, why a<->b is true if both a and b are false is also a difficult concept to grasp.
\n" ); document.write( "The recommendation is to follow the rules regardless of whether you fully understand the concept behind them.
\n" ); document.write( "That's what I did above.
\n" ); document.write( "Hopefully I did it right.
\n" ); document.write( "Here's a reference on conditional statements that might help you to understand better.
\n" ); document.write( "http://www.rwc.uc.edu/koehler/comath/21.html
\n" ); document.write( "scroll down to IMPLIES and EQUIVALENCE you will see their truth tables.
\n" ); document.write( "The IMPLIES truth table is true in all cases except the case where the first variable is true and the second variable is false.
\n" ); document.write( "In that case, the implies statement is false.
\n" ); document.write( "It is true if both variable are true.
\n" ); document.write( "It is also true if the first variable is false regardless of the value of the second variable.
\n" ); document.write( "In the IMPLIES statement, the first variable is called the precedent and the second variable is called the conclusion.
\n" ); document.write( "The EQUIVALENCE truth table is true only when the first variable and the second variable agree.
\n" ); document.write( "They can both be true or they can both be false.
\n" ); document.write( "If they do not agree, i.e. one is true and the other is false, then the EQUIVALENCE statement is false.
\n" ); document.write( "The EQUIVALENCE STATEMENT is equivalent to (no pun intended) a->b AND b->a
\n" ); document.write( "This would show up as:
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\n" ); document.write( "You have to buy the logic in the implies statement in order to buy the logic in the equivalence statement.
\n" ); document.write( "Once you buy the logic in the implies statement, than the logic in the equivalence statement can be shown through the use of the truth tables.
\n" ); document.write( "Here's a truth table showing you the relationship between the two.
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document.write( "      a       b       a->b      b->a      a->b^b->a        a<->b\r\n" );
document.write( "      T       T         T         T           T              T\r\n" );
document.write( "      T       F         F         T           F              F\r\n" );
document.write( "      F       T         T         F           F              F\r\n" );
document.write( "      F       F         T         T           T              T\r\n" );
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\n" ); document.write( "Remember that a->b is true if a is false regardless of the value of b.
\n" ); document.write( "Remember that b->a is true if b is false regardless of the value of a.
\n" ); document.write( "The only time a->b is false is if a is true and b is false.
\n" ); document.write( "The only time b->a is false is if b is true and a is false.
\n" ); document.write( "This truth table shows that a<->b is equivalent to a->b^b->a\r
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