In mathematics the study of logic deals with statements or propositions. A statement is a sentence that is either true or false, but not both. The statement is said to have a truth value.
“All children are lefthanded” is a false statement. “Many people enjoy science fiction” is a true statement. “5 + 4 = 9” is a true statement while 5+4=8 is a false statement. “This sentence is true” is not a statement at all since it can be both true and false. The same is true for “6+n=9.” It is neither a true nor false statement since this depends upon the replacement for the variable n.
In logic we use symbols to represent statements. For example, if we use p to represent a statement then we use ~ p to represent the negation of p. If p is “I am a plumber” then the negation of p written, ~ p, would be “I am not a plumber.” It is worth noting that the negation of p must be false if p is true and must be true when p is false. Try the following:
Part 1 Identify each of the following as a statement or not a statement.
a. The house is red.
b. Joan is not at home.
c. n+3 = 7 d. 6+2 = 7
Part 2 Give the negation for each of the following statements.
a. I like school
b. No one is happy at work.
c. All trees are green.
Part 1. a. yes b. yes c. no d. yes.
Part 2. a. I do not like school b. Some people are happy at work c. Not all trees are green.
Note In example part 2b the negation cannot be all people are happy at work since it is possible for some people to be happy at work and some to be unhappy at work. Some statements are a little harder to negate, especially when they include all or none statements.