The process of negating compound statements uses the tilde symbol,~~, which we have already used in negating simple statements. It is important to note that in negating compound statements the tilde symbol is placed before the statement we wish to negate. If we wish to negate the entire compound statement we place the tilde before a parenthesis. Example.
In p ^q only the p statement is negated while in ~ (p ^ q) the whole statement is negated.
To negate a conjunction or disjunction of two or more simple statements we negate each of the simple statements and then change all of the connectives from disjunction’s to conjunctions and all of the conjunctions to disjunction’s.
~ (~P q) = p^~ q
Try these. Give the negation for each of the following:
2. ~ (P^ q)
1. pvr~ q 2.) ~ p ^~ q.
The reason we are able to negate each simple statement and change the connectives is that the statements are equivalent. In the example above if we let p be “I try” and q be “I succeed” then the negation of I do not try or I succeed written ~( ~ p ^q) would be “I try” and “I do not succeed” written p^~q. Both compound statements say the same thing in different ways are equivalent.
The teacher may wish to give the class exercises such as those which follow: Let p be I race and let q be I win. Give a verbal translation for each of the following:
a. P^ q
b. p^~ q
c. ~ p ^~ q
e. (P v~ q)
f. (p^~ q) v( ~ p ^~ q)
a. I race and I win
b. I race and I do not win
c. I do not race and I do not win
d. It is not true that I race or that I do not win
e.’ The statement I do not race or I do not win is not true.
f. I race and I do not win or I do not race and I do not win.
Note: In solution d. the language becomes very difficult and so we may wish to use the equivalent statement, “I do not race and I win.” We know this is equivalent since in negating any compound statement we can simply negate each simple statement and change all of the connectives. In this example for instance the two equivalent statements would be written symbolically as follows: ~(pv ~ q)=p^q.
Note: = is the symbol for equivalent.
A truth table for the above problem will be constructed at the end of the section dealing with truth tables.