Quantifiers

This post belongs to my “Intro Math” series, a set of posts designed to give the foundations to advanced mathematics. More info can be found in Intro Math.

Before reading, I recommend Set Theory Part 1 for a brief overview of sets.

In mathematics, there are two main quantifiers and . These symbols are very important for understanding definitions and theorems.

First, let’s look at . This symbol is read as for all. An example statement using it is . This would be read as “for all x in X, x is greater than or equal to 2.” Essentially, this statement is telling us about elements of .

Example 1 – The set does not satisfy the condition above since .

Next, we’ll look at . This symbol reads as there exists.An example statement using it is . This would be read as “there exists an x in X such that x is greater than or equal to two.” The symbol stands for “such that.” The statement just requires at least one element of X to be greater than 2.

Example 2 – The set does satisfy the condition above since there is at least one element that works.

Notice that it is important to include the set an element is from. The meaning of these examples could change depending on the set chosen.

These quantifiers can be combined to create much more interesting statements. Consider the condition . This statements reads “for all x in X, there exists a y in Y such that x > y.” To check this statement, we need to check each element in X, and find a y that satisfies x > y. The y does not have to be the same for each x chosen.

Example 3 – Define . These sets do not satisfy the above condition, consider , there is no element of y where x > y.

Flipping the quantifiers can drastically change the meaning of a statement. Define , which reads as “there is an x in X such that for all y in Y, x > y.”

Example 4 – Define , the same as Example 3. These sets do satisfy this new condition. Consider , which is greater than every element of Y.

Often when dealing with mathematical statements, it is important to consider the negation. How do we negate qualifiers? To negate , there must be at least one element which does not satisfy the condition.

Example 5 – Consider the statement . The negation of this is .

Similarly, to negate , all elements must not satisfy the statement.

In general, to negate a statement with quantifiers, you flip each of the quantifiers then replace the statement at the end with the opposite.

Example 6 – Let’s find the opposite of used above. Flipping the quantifiers and negating the last statement gives .

As with all of mathematics, it’s important to carefully think through each definition rather than mechanically working. Make sure you understand why the negations shown above are valid.

There are a few additional symbols that can appear when dealing with quantifiers. The symbol reads as “there does not exists” and is an alternate negation for . I prefer the earlier method, as it stays consistent with the negation of .

Another alternate for “there exists” is , which reads as “there exists a unique.” For this to be satisfied, there must be exactly one element satisfying the condition.

Mastering these quantifiers is critical to learning mathematics. It always helps to work through each definition carefully.

Problems

1.) Write out the statements in Example 1-6 as words, similar to what I did when I introduced each quantifier.

2.) Negate the following statements.

a. .

b. .

c. . There are two ways to do this.

d. .

3.) Review the definition of “subset” and “bridge” in Set Theory Part 1. Write negations for both.

If you’ve read this post and Logical Implications, you have a good basis for understanding symbolic definitions. Test your skills by reading Bijective Functions and working through the definitions provided.