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## Introduction to :

**Mathematical statements** are exactly the same as fashion statements. Except instead of clothes, we have mathematical formulas. Hopefully we won't get chilly walking down the runway.

The simplest kind of mathematical statement is an explanation of how numbers are related. For example, you might say, "*x* = 5" or, "4 + 7 = 35" or, "58 is the sum of two prime numbers." As you can see, some statements are true, some are false, and some are as clear as a mud smoothie.

What all of these statements have in common is that they can't be split into simpler statements—they are *indivisible* (with liberty and justice for all). The Greek word for indivisible is *atomos*, so we call these statements **atoms**. Unlike the atoms in chemistry, mathematical atoms can make only statements, not bombs.

#### Example 1

Does any value of *s* make the statement "*s* > 5 and *s* < 2" true? | |

For the statement to be correct, we need to satisfy both parts. Unfortunately, we can't have a number that's both greater than five and less than two. That means no value of *s* will make that statement true. | |

#### Example 2

Does any food satisfy, "This food is a pie or it tastes like a blueberry"? | |

Plenty of foods do. Since the two atoms in the statement are joined by a disjunction, only one part of the overall statement has to be true to make the whole thing true. For "this food is a pie" we could have the food be any kind of pie (not π). For the second statement, any food that tastes like a blueberry would also work, from blueberry smoothies or ice cream or blueberries themselves. Blueberry pie would satisfy both parts of the statement, so the overall statement would still be true. | |

#### Example 3

Is there any value of *x* that makes "*x* < 4 *and* not(*x*^{2} < 20)" true? | |

A conjunction means each part must be true for the whole statement to be true. Let's break it down into atoms again. Lots of numbers are less than 4. Next. Not(*x*^{2} < 20) is a negation of whatever is inside the parentheses. So the negation of "*x*^{2} is less than 20" would be "*x*^{2} is not less than 20" or "*x*^{2} is greater than or equal to 20." For the entire statement to be true, we need a number that is less than four but greater than 20 when squared. Does such a number exist? You betcha. If we make *x* = -5, it satisfies both atoms in the statement, and therefore makes the whole statement true. | |

#### Exercise 1

Sandra loves skydiving, pepperoni pizza, and the number 17. She doesn't like Tuesdays or the color purple (sorry, Alice Walker). Is the statement "Sandra loves pepperoni pizza and she loves Tuesdays" true?

Hint

For the statement to be true, both parts have to be true.

Answer

No, the statement is false.

#### Exercise 2

Sandra loves skydiving, pepperoni pizza, and the number 17. She doesn't like Tuesdays or the color purple (sorry, Alice Walker). Is the statement "Sandra loves 17 or she does not love 17" true?

Hint

Only one part of the overall statement can be true. Does that make it true or false?

Answer

The statement is true.

#### Exercise 3

Sandra loves skydiving, pepperoni pizza, and the number 17. She doesn't like Tuesdays or the color purple (sorry, Alice Walker). Is the statement "Sandra does not love purple or she loves skydiving" true?

Hint

What does it mean if both are true?

Answer

The statement is true.

#### Exercise 4

Is there a value for *x* that will satisfy the statement "*x* = 7 or *x*^{2} = 25"?

Hint

"Or" means only one of the atoms must be true for the whole statement to be true.

Answer

Yes. There are three values: *x* can equal either 7 or 5 or -5.

#### Exercise 5

Is there a value that will satisfy the statement "*x*^{0} = 1 and *x* + 1 = 4,623"?

Hint

Both atoms must be true for the statement as a whole to be true.

Answer

Yes. The value that will satisfy the whole statement is *x* = 4,622.