Notation PDF

Some recurring notation Definition 1 A set is a collection of objects, called members of the set. Two sets A, B are equal if they contain exactly the same elements; in this case, we write A = B. There are two ways we can define a set: (i) by li...

Some recurring notation Definition 1 A set is a collection of objects, called members of the set. Two sets A, B are equal if they contain exactly the same elements; in this case, we write A = B. There are two ways we can define a set: (i) by listing its elements inside curly brackets { }: Example A = {−1, 0, 1} is the set containing the elements −1, 0, 1. Note that the order does not matter, so A = {0, 1, −1}, A = {1, −1, 0} etc. (ii) by a rule: {x : rule} means the set of all x that satisfy the given rule (: means “such that”). Example A = {x : x is an integer and − 1 ≤ x ≤ 1} (to be read “A is the set of all x such that x is an integer and −1 ≤ x ≤ 1). Then A is equal to {−1, 0, 1}. Example B = {x : x is a first year BME Bachelor student and x is at least 1.70m tall}. This notation is particularly convenient when the set has infinitely many members (see examples below). We write x ∈ A if x is a member of the set A and x ̸∈ A otherwise. For sets A, B we write A ⊂ B if every element of A is an element of B and A ̸⊂ B if there is at least one element of A that is not in B. If A ⊂ B, we say that A is a subset of B. Example For A as in Example 1, we have −1 ∈ A, but 2 ̸∈ A. Furthermore, {−1, 1} ⊂ A, but {5, 1} ̸⊂ A. Some sets of numbers that we will consider are the set of natural numbers N = {x : x is a positive integer } = {1, 2,... } the set of integers Z = {x : x is an integer } = {· · · − 2, −1, 0, 1, 2,... } the set of real numbers R. the set of complex numbers C = {x + iy : x, y ∈ R} We have N ⊂ Z ⊂ R ⊂ C; i ∈ C, i ̸∈ R. The set containing no elements is called the empty set and is denoted by ∅. Finally, when performing mathematical steps, we are often using the following ideas. if Statement (1) holds, then Statement (2) holds. We write (1) ⇒ (2) (to read “(1) implies (2)”). Example x = 1 ⇒ x2 = 1 (since, if x = 1, then x2 = (1)2 = 1). Statement (1) holds if Statement (2) holds. We write (1) ⇐ (2) (to read “(1) is implied by (2)”). 1  If both (1) ⇒ (2) and (1) ⇐ (2) hold, we write (1) ⇔ (2) and say (1) holds if and only if (2) holds. Example (−1)2 = 1, so we cannot replace ⇒ with ⇔ in x = 1 ⇒ x2 = 1. That is to say, if x2 = 1, then we cannot conclude that x = 1. However, we do have x2 = 1 ⇔ x ∈ {−1, 1}. So in summary this is the notation we have introduced so far: : such that {} set notation ∈ is in/belongs to ∈ / is not in/does not belong to ⊂ is a subset of ̸ ⊂ is not a subset of ∀ for all ∃ there exists ⇒ implies/then ⇐ is implied by/if ⇔ implies and is implied by/if and only if 2