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Lesson 144: Mathematical Logic Using Sentences
In this lesson we'll begin the study of mathematical logic. First we'll learn the material using real English sentences, and in the next lesson we'll use symbols. Note that not all schools teach this topic. Find out if your school does, and if you will be tested on it.
The lesson has a lot of definitions which need to be memorized. In math, we define a statement as a sentence that is true or false, but not both. Examples would be "seven is more than three," or "George Bush is the President of China." Of course the first is true, and the second is false.
An open sentence as a sentence that has at least one variable. Examples would be "n is less than 17" or "two numbers add up to 12."
A negation is the opposite of a statement. Usually, we just add the word "not." For example, the negation of "I own a car" is "I do not own a car".
A conjunction is a compound statement that is formed by joining two statements with the word "and." It doesn't matter if the statements are true or false, the joining will still be considered a conjunction. An example is, "My name is Larry, and I play the piano."
A disjunction is a compound statement that is formed by joining two statements with the word "or." It doesn't matter if the statements are true or false, the joining will still be considered a disjunction. An example is, "My name is Billy, or today is Monday."
A tautology is a compound statement that is always true. A common example is of the form: "It is raining, or it is not raining."
A conditional is a compound statement that is formed by joining two statements with the words "if...then." An example is, "If it snows, then I will go skiing".
There are three possible variations of every conditional statement. Let's start with the sample conditional above: "If it snows, then I will go skiing".
The converse would be: "If I go skiing,
then it is snowing"
The inverse would be: "If it does not snow,
then I will not go skiing"
The contrapositive would be: "If I do not go
skiing, then it is not snowing"
These will be much easier to memorize and visualize when we represent these statements using symbols in the next lesson. For now, just notice what has been done with the conditional in each of the three variations.
A biconditional is a statement that is formed by joining two statements with the words "if and only if". An example would be, "I will buy the book if and only if it is on sale." Sometimes "if and only if" is abbreviated as "iff" ("if" with an extra f).
Much of this lesson will make more sense after studying the next lesson, in which we use special symbols and truth tables to work with sentences such as these.
Remember that you can ask a math question if you have additional questions about a topic, or you can contact me if you have any comments or suggestions for this site.