Counting the Real Numbers

If we are in a room full of ballroom dancers where each male dancer has a female dancer partner, and no one is left without a partner, we can say that there are as many male as female dancers in the room even without counting. In mathematics, we say that there is a one-to-one correspondence between the set of male dancers and the set of female dancers.

Pairing Infinite Sets

In the A Glimpse at Infinite Sets, we have learned that if we can pair two sets in one-to-one correspondence, we can say that the two sets have the same number of elements. The number of elements of a set is its cardinality. Therefore, the cardinality of the binary numbers {1,0} is 2 and the cardinality of the set of the vowel letters in the English alphabet {a, e, i, o, u} is 5.

The pairing of sets can be extended to compare sets with infinite number of elements or infinite sets.  In Figure 1, it is clear that it is possible to pair the set of integers with the set of counting numbers in one-to-one correspondence (can you see why?).  Infinite sets whose elements can be paired with the set of counting numbers in one-to-one correspondence is said to be countably infinite.

Figure 1

As a consequence of the analogy above, we can conclude the cardinality of counting numbers is equal to the cardinality of integers (Can you see why?). » Read more

On the Manila Hostage Incident

I would like to personally apologize to the family, kins and friends of the Hong Kong nationals in last Monday’s hostage incident in Manila.  It has really taken a lot out of me in the past week.

We know our apologies would never be enough to alleviate or take away the pain that our countryman has caused you, but we want to say it anyway. Many of us are as sad and as disappointed as you.

My sincerest apologies and deepest condolences.

To my fellow Filipinos:

We will live our life only once.  I hope that we live it for the benefit of others, for the honor of our country, and for the glory of God.

GeoGebra Tutorial 32 – Graphing Piecewise Functions

In this tutorial, we are going to learn how to graph piecewise functions. In our example, we will graph the piecewise function

f(x) = \begin{cases} 1-x, x \leq 1 \\x^2, x > 1 \end{cases}.

The output of our tutorial is shown below.  GeoGebra has not yet developed a way to construct piecewise functions, however, its features is more than good enough for improvisation. In the construction below, we will use the function command to graph functions with specified domains. We will create manually, the two endpoints of the functions on (1,1) and (1,0). We will also use the vector tool to construct arrows.

Click the image to go to view the applet.

If you want to follow this tutorial step-by-step, you can open the GeoGebra window in your browser by clicking here. The output applet of this tutorial can be viewed here. » Read more

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