Understanding the Basics of the De Morgan’s Laws

In the previous two posts, we have discussed about the logical operators and, or, and not.  Those two articles are preparation for this post.

Consider the following inequalities.

1.) x \leq 8

2.) -2 < x < 3 

The first inequality can also be represented as x < 8 or x = 8

Now, how do we find

not (x < 8 or x = 8).

To easily understand the question, we graph x \leq 8 first and then see what’s not on that graph.  » Read more

One to One Correspondence and Hilbert’s Grand Hotel

In the Understanding Hilbert’s Grand Hotel, we have discussed the brilliant schemes of a hotel manager in accommodating finite and infinite number of guests in a hotel with infinite number of rooms, where each room was occupied by one guest. In other words, the hotel was fully occupied. In this post, I will explain the mathematics behind these schemes. To be able to understand the explanation, it his highly recommended that you read first the post in the link above.

Finite Number Of Guests

In the Grand Hotel problem, during the first night, a guest arrived. The hotel was full, so there was no room available. However, to accommodate the new guest, the manager requested the guest in Room 1 to move to Room 2, the guest in Room 2 to move to Room 3, the guest in Room 3 to move to room 4 and so on. This means that each guest had to move to the room whose number is 1 higher than the the current room number. This leaves the Room 1 vacant.

Now, how is this possible? » Read more

The Number of Points on Two Line Segments

We say that a set is countably infinite if we can pair the elements with set of counting numbers 1, 2, 3, and so on. Believe it or not, the number of positive integers and the number of integers (both negative and positive including 0) have the same number of elements. It is because we can pair them in a one-to-one correspondence such as shown in the below.

As shown on the table, if we continue indefinitely, we know that we can pair each counting number with an integer in a one-to-one correspondence without missing any element.

counting numbers integers bijection

Using this concept, we show intuitively that the number of points on two line segments is equal even if they have different lengths. We can do this by showing that for each point on segment \overline{AB}, there is a corresponding point on segment \overline{CD}. » Read more

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