# Logic and Proof

### Topics

## Introduction to :

Understanding **equality**, or sameness, is a universal theme in all areas of mathematics. When mathematicians say, "2 + 2 = 4," they mean that the two things on either side of the equals sign are literally the same thing: they are two ways of describing the same object. They are equal.

In mathematics, the objects we work with are usually numbers. The thing is, we can't usually see them (talk about imaginary numbers, huh?), and we're forced to make do with looking at their descriptions instead.

Somebody with too much free time might choose to write "(1 + 1 + 1 + 1) – (1 + 1 + 1) + 1 + (1 – 1 – 1) – (1 – 1) + (1 + 1)" instead of the significantly easier to read "3," even though they describe the exact same number. Way to mess with us.

#### Example 1

Identify the property exemplified here. Turkey Sandwich = Turkey Sandwich. |

#### Example 2

We're given that |

#### Example 3

If |

#### Example 4

If |

#### Exercise 1

Give an example of the symmetrical property.

#### Exercise 2

If we're given a puppy, what can we say about the puppy and which property would we be using?

#### Exercise 3

^{A}⁄_{B} = ^{C}⁄_{D}. Which statements would make this true?

#### Exercise 4

*X* = *Y* and *Y* = 3. Which properties allow or forbid us to say that 3 = *X*?

#### Exercise 5

*A* = *B* and *X* = *Y*. Which properties allow or forbid us to say that ?

#### Exercise 6

*A* = *B*, *B* = *C*, *C* = *D*, and *D* = *E*, *E* = *F*, and *F* = *G*. Which properties allow or forbid us to say that *A*(*B* + *C*) = *D* × *E* + *F* × *G*?