In the last section, we learned how to negate a conjunction and a disjunction using DeMorgan’s Laws. Here, we will also learn how to negate the conditional and quantified statements.
Negating an Implication
Recall that we learned about implications. Implications are logical conditional sentences stating that a statement p, called the antecedent, implies a consequence q. Also recall that the truth table for an implication has three trues and one false in the final column. The only place where the implication was false was when the antecedent is true and the consequent is false. So to negate an implication, we want that case to be true and all others to be false. So the negation of an implication is p ∧ ~q.
Key Takeaways
The negation of p→q is p ∧~q.
Or in symbols, ~ (p→q) = p ∧~q.
Quantified Statements
We use quantified statements often in mathematics, but they are also used in everyday life. An example of a quantified statement in math might be “All even numbers are divisible by 2.” An example of a non-mathematical quantified statement might be “On every assignment, there is someone who forgets to complete the assignment.”
In math, we use two symbols for quantified statements: the universal quantifier and the existential quantifier. The universal quantifier means “for all” or “for every”. It is denoted by ∀. The existential quantifier means “there exists” or “for some”. It is denoted by ∃.
Examples
Determine if each statement uses the universal or existential quantifier.
- All mathematicians wear glasses.
- There exists some number such that x2 = x.
- All politicians are dishonest.
Negating Quantified Statements
Recall that negating a statement changes its truth value. So negating a true statement gives a false statement and negating a false statement gives a true statement.
Key Takeaways
The negation of a universal statement (All A are B) is an existential statement (There exists some A that is not B).
The negation of an existential statement (There exists some A that is B) is a universal statement (All A are not B).
Examples
Negate each statement.
- All mathematicians wear glasses.
- There exists some number such that x2 = x.
- All politicians are dishonest.
1: There exists at least one mathematician that does not wear glasses.
This is a universal statement. So the negation will be an existential statement. Notice that the original statement was false, and the negation is true.
2: For all numbers, x2 ≠ x.
This is an existential statement, so the negation will be a universal statement. Notice that the original statement is true (0 and 1 fit the property), and the negation is false.
3: There exists at least one politician that is honest.
This is a universal statement. So the negation will be an existential statement.
