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 x1 and x2 in the interval, if x1 < x2 then f(x1) < f(x2).” 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.