New terms in this lesson:

and, or, truth table, propositional expression, truth value

In Boolean Logic, A is either true *or* false. What is *or*? What is *and*?

What do you mean when you say *and*? What do you mean when you say *or*?

Can you come up with some alternative definitions of and and or?

*and* � additionally, both, all, as well as

*or* � either, optional, one of

What do *and* and *or* actually mean? Mathematicians cannot stand the multiple definitions of *and* and *or* because they cause ambiguity. Instead, they defined unique, very specific definitions for and and or. These mathematical, carefully defined words are somewhat different from our everyday English *or* or *and*, so mathematicians use a unique symbol to represent them. The the symbol represents *or*. So 4 *or* -4 would be written 4 -4.

Similarly, mathematicians use to represent *and*. 4 *and* -4 would be written 4 -4.

So what do the mathematical *or* and *and* mean? If statement A *and* statement B is true, this means that **both** statements must be true. If statement A *or* statement B is true, then at least **one of them** must be true.

These definitions aren’t really very different from the everyday forms of *and* and *or* you are already familiar with. Imagine you ordered a jacket off the internet. The seller could not guarantee you a particular color, but the website said it would be either “blue *or* green.� Imagine your surprise, then, when you open the shipping box you pull out a red jacket. The promise from the seller was false, because neither of the statements was true; the jacket was not blue or green. In order for an *or* expression to be true at least one of the statements must be true.

Now imagine you’re shoe shopping online and you find a pair of sneakers you like that are black and red. The website says that the shoes you will receive are the same as the ones pictured, but when your pair arrives they are solid black! The promise from the seller was, once again, false, this time because only one of the statements was true. The shoes are black, but not red. In order for an *and* expression to be true both of the statements have to be true.

Expressions using *and* and *or* can get very complicated so to help us evaluate them we can use a __truth table__. A truth table is an organized way to evaluate a set of statements using different givens. It allows you to experiment with different assumptions to see which one works best and to see if an expression is true with a variety of givens or only under certain conditions.

Truth tables use columns to represent the pieces of an expression and rows to represent the different starting assumptions that can lead to different conclusions.

Look at the truth table below to evaluate the expression **A and B** or **A B**. First you will see that we have a column for each statement � in this case a column for statement **A** and a column for statement **B**. We also have a column for the __propositional expression__ **A and B**. A propositional expression is a statement connected with

In creating the truth table we fill in the boxes below each statement with a series of true and false. There are four possible true/false combinations for two assumptions: A and B can both be true, only A could be true, only B could be true, or A and B could both be false. Our table needs four rows to represent these four cases. We then look to these to evaluate the expression **A and B**. For *and*, our rule is that if __both__ statements are __true__, then our expression is __true__.

A | B | A and B |
---|---|---|

True | True | True and True = True |

True | False | True and False = False |

False | True | False and True = False |

False | False | False and False = False |

Looking at our truth table we can see that the propositional expression “A and B� was only true in one case. For the expression to be true we have to assume that both A and B are true.

A | B | A or B |
---|---|---|

True | True | True or True = True |

True | False | True or False =True |

False | True | False or True = |

False | False | False or False = |

Below is a truth table for *or*. Again we fill in the true and false pattern below the statements and evaluate the expression **A or B**.

You can see that there is a big difference between the and truth table *and* the *or* truth table. Whereas for *and*, both statements had to be true, just one of them had to be true for *or*. That is the difference between one true scenario and three. That is significant!

In everyday life we are constantly choosing things using *and* and *or*. Have you ever said, “I want the white *and* blue shoes, *or* the black boots,� or something similar? You are using *and* and *or* in very powerful ways. Because these words connect two or more ideas, they are also called *connectives*. When you use *and* and *or* in a mathematical sense, you have to be very careful!

Thinking back to our last lesson, we worked with the given |x| = 4. Let’s look at the differences between [x = 4 or x = -4] and [x = 4 and -4]. Which one of these expressions is true for sure, without any doubt?

Thinking back to our last lesson, we worked with the given |x| = 4. Let’s look at the differences between [x = 4 or x = -4] and [x = 4 and -4]. Which one of these expressions is true for sure, without any doubt?

Look at [x = 4 or x = -4]. Let’s think about some things we know about this statement. The two cases do not have to be true at the same time. At least one of them is true. They are not both false, although one of them has to be false because x cannot be both 4 and -4 simultaneously! This is the only answer a mathematician would consider a true for sure statement.

Now look at [x = 4 and x = -4.] Would you say this expression is true, without a doubt? Hopefully not! It looks like it could make sense, but from a mathematical standpoint it does not work because both would have to be true at the same time due to the use of *and*. This is not correct because x can only equal one of the values at a time. Only one of the two can be true. That’s why we use *or* to distinguish one from two or more values.

The *and* and *or* symbols are often used in combination with the *not* symbol, as in *not* A. This means the opposite of A. Remember we introduced this symbol in lesson 1.

Mathematicians use the symbol � to represent *not*. So an expression could read like this: [A � B]. Using words you would say it: A *and not* B.

Using our new symbols, how would you write A *or not* B?

Let’s use our new symbol � in a truth table. We want to complete the table for the expression [A � B]. For this truth table we need at least three columns: one for each of the statements (A and B) and one for the propositional expression. However, it is in our best interest to add one more column to simplify our evaluation. Within our longer expression is the smaller expression “¬ B�. We give this its own column so that we can evaluate it first, then do the longer expression.

A | B | � B (opposite of B) | A � B |
---|---|---|---|

True | True | False | True and False = False |

True | False | True | True and True = True |

False | True | False | False and False = False |

False | False | True | False and True = False |

__Evaluating Propositional Expressions with Multiple Components __

With a good understanding of the differences between and , we can make expressions longer and more complex. The truth table grows along with the expression. As the expressions get more complicated you will need to add columns, as the assumptions increase in number you will need to add rows.

Let’s make a truth table for A B A (A *and* B *or* A). The easiest way to solve a multipart expression is to break it down so that you are never solving for more than two statements at a time. To start with, let’s ignore the end of the expression and just work on the beginning. A B is our first part.

Before we solve for this expression, what is the rule for ? Remember that __both__ statements have to be __true__ for the expression to be __true__.

A | B | A B | A B A |
---|---|---|---|

True | True | True and True = True | |

True | False | True and False = False | |

False | True | False and True = False | |

False | False | False and False = False |

We have solved the first part of the expression; this first part of the expression now acts as a single unit, a single statement, because we have evaluated it for a true or false answer. We effectively are now evaluating for [A B] A, so we are still only working with two statements at a time.

Before we solve for the expression A B A, let’s review our rules . Remember for to be true, at least one statement must be true. Can you fill in the table? You need to compare the condition of [A B] and the condition of A and remember that you are evaluating for *or*, .

A | B | A B | A B A |
---|---|---|---|

True | True | True and True = True | True and True = |

True | False | True and False = False | False and True = |

False | True | False and True = False | False and False = |

False | False | False and False = False | False and False = |

A. Sense from Nonsensei - *Answers may vary*

Punctuate the following fragments so that they make sense:

- It was and I said not or
- Picture this a grey squirrel deep in the forest nibbles on the yummies of its environment fruits seeds anything fresh and pluckable

- CyberProf(TM)