January 2012 - Page 5 of 9 - Math and Multimedia

Proof by Contradiction: Knights and Liars

January 19, 2012 GB College Mathematics, High School Mathematics

I am currently reading a local book about mathematical problem solving and I came across with a cute proof demonstrating proof by contradiction. Actually, I heard a very similar problem ages ago, but I never really thought that it was mathematical and never really bothered to solve it.

Anyway, according to the authors, the problem was based on the lectures given by Oleg Goldberg (an International Math Olympiad gold medalist) at the IDEA MATH program, a weekend extracurricular training on problem solving [in Massachussettes?]. The problem and proof are as follows.

Problem

All natives of the Idea island are either knights or liars. Knights always tell the truth, and liars always lie. A person says “I am a liar.” Prove that he is not an Idea native.

Note: The idea of putting the image below is to let you do the proof first before scrolling down. So, no peeking please. » Read more

3 comments knights and liars, mathematical proofs, methods of proof, proof by contradiction, proof by contradiction example

Sets: Terminologies, Notations, and Operations

January 18, 2012 GB High School Mathematics, Set Theory

As a preparation for more posts on probability, statistics, permutations and combinations, we familiarized ourselves last week with the different terminologies and notations of probability. We continue in this post by studying set terminologies, notations, and operations. Note that this is also the third post in the Set Primer Series; the first and second are Introduction to Sets and Subset: a set contained in a set.

Universal Set

The universal set is the set that contains all the elements under discussion. If we talk about the letters in the English alphabet, then the universal set contains all the 26 letters. In set theory, universal set is usually denoted by $U$ .

In the following discussion, we let $U$ be the set of integers, $E$ be the set of even integers, and $O$ be the set of odd integers. The following are the common operations on sets.

Intersection

If sets $A$ and $B$ have elements in common they form a set written as $A \cap B$ . This is the intersection of $A$ and $B$ .

Intersection of Sets

Example: If we let $A = \{1, 2, 3, 4, 5\}$ and $B = \{2, 4, 6\}$ then $A \cap B = \{2,4\}$ . » Read more

One comment intersection of sets, sets, union of sets, universal set

Carnival 19 Deadline of Submission

January 17, 2012 GB Blog Carnival

It’s carnival time once more and the Mathematics and Multimedia Carnival is home. If you have a blog post about school mathematics, multimedia, mathematics teaching, math games and puzzles, etc., you may submit here or submit directly to mathandmultimedia@gmail.com. The 19th edition of the Math and Multimedia Carnival will be posted here on January 23, 2012.

Here are last month’s carnivals

For those who are not familiar with math carnivals, you may want to read “What is a blog carnival?“

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