Elementary Functions and Calculus I Math 131 (Sec 42), Autumn 2004 Handout 1

• As was talked about in class, a set is a collection of objects. If a set A contains the objects a, b and c, we write A = {a, b, c}. We usually will talk about sets of numbers, e.g. N = {1, 2, 3, 4, …}. Can you think of more examples?

• We will write a ∈ A to mean that a is an element of (or is contained in) A. For example 1 ∈ N.

• In our first class we mostly talked about things we will assume (e.g. x + y = y + x for any real numbers x and y). These things are called axioms. From the axioms we will derive or prove results. These results are deduced using only the axioms and logic. Provided our axioms are true the results will be too. We often call important results theorems.

• Theorems often take the form If A, then B, where A and B are statements. For example If x ≠ 0, then x^{2} > 0. We use the notation A ⇒ B to mean If A, then B. Similarly, we use the notation A ⇐ B to mean A only if B. If A ⇒ B and A ⇐ B we write A ⇔ B.

• For a statement A we write ∼ A to mean the negation of A. For example, if A is the statement Paul has no apples then ∼ A is the statement Paul has some apples. We have that (i) A ⇒ B is equivalent to the statement (ii) ∼ A ⇐∼ B. We call (ii) the contrapositive of (i).

• Since some phases occur often in mathematics we use some useful abreviations. The symbol ∀ means for all or for any. The symbol ∃ means there exists and such that is written s.t.. For example there exists areal number x such that 5x = 1 is written ∃x ∈ R s.t. 5x = 1.

• If you are confused about anything on this sheet it may help to read through A bit of logic on page 3 of the text and Quantifiers on page 4. If this doesn’t help please ask your tutor or the lecturer.