**The universal quantifier – for any – for every – for all**

Many theorems and definitions in mathematics use the phrases “for any”, “for every” or “for all.” The upside-down A is the symbol. The three phrases all mean the same thing!

For example, we have the definition “A function is increasing on an interval if, and only if, for all pairs of numbers *x*_{1} and *x*_{2} in the interval, if *x*_{1 }< *x*_{2} then *f*(*x*_{1}) < *f*(*x*_{2}).” Whenever you have a theorem or definition with one, restating it with the other two will help students understand it better: “for *all pairs* of numbers,” “for *any pair* of numbers” and “for *every pair* of numbers.”

**Increasing and Decreasing Functions**

The symbols in the definition above tell the whole story – sure they do. As with any theorem or definition, use the Rule of Four. The definition above is the analytic part. The graphical part is the obvious – the graph goes up to the right. The numerical part is that as the *x*-values increase in a table, so do the *y*-values. The verbal part is the two preceding sentences and all the talking you’re going to have to do to explain this.

The function increases on the closed interval and the function decreases on the closed interval . The fact that is in both intervals is not a problem since it is in the intervals, not at the point, that the function increases or decreases. This is because is larger than all (every, any) values in , and also larger than all (any, every) of the values in .

“Playing” with theorems: You will soon have a theorem that says, “If the derivative of a function is positive on an interval, then the function is increasing on the interval.” Nothing in the paragraph above contradicts this, because the hypothesis says nothing about what is true if the derivative is zero. For this you have to go back to the definition. The converse of this theorem is false. Counterexample: is increasing on any (all, every) interval containing the origin, yet . The AP exams do not make a big deal of this; they accept either open or closed intervals for increasing or decreasing.

Readers know most of the common abbreviations and can usually figure out the others as long as they are not too bizarre. Certainly they will know the backwards E, the upside down A, the vertical bar, “b/c” and the three dots in a triangle. Students will not lose credit for this.

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Hi Lin,

Are these symbols accepted by AP Test readers? I use abbreviations for “therefore” and “such that” and “for every” in class but have been told to discourage students from using them. Any thoughts?

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