Review

Use these notes to review Chapter 1 quickly.

Core Ideas

Use each idea as a reading move:

• Notation: compressed language. Can you expand it into words?
• Variable: name for a quantity. Can you say what value or object it names?
• Expression: recipe for computing a value. Can you substitute a small input?
• Function: reusable input-output rule. Can you identify input, rule, and output?
• Set: collection of distinct objects. Can you test membership?
• Interval: continuous number range. Can you tell which endpoints belong?
• Tuple: ordered collection. Can you explain why order matters?
• Index: position selector. Can you tell whether indexing starts at 0 or 1?
• Summation: repeated addition. Can you expand it for three terms?
• Product: repeated multiplication. Can you expand it for three terms?

Key Notation

Read notation in complete phrases:

• $x$: "x"; a named quantity.
• $f(x)$: "f of x"; the output of function $f$ at input $x$.
• $a \in A$: "a is in A"; membership in a set.
• $[a,b)$: numbers from $a$ up to $b$; include $a$ and exclude $b$.
• $x_i$: "x sub i"; the i-th item of $x$.
• $\sum_{i=1}^{n} x_i$: "sum x sub i from i equals 1 to n"; add the items.
• $\prod_{i=1}^{n} x_i$: "product of x sub i from i equals 1 to n"; multiply the items.
• $\frac{1}{n}\sum_{i=1}^{n}\ell_i$: "one over n times the sum of ell sub i from i equals 1 to n"; average the losses.

Greek letters are names too. Saying "theta" or "lambda" is only the first step; the important question is what that symbol represents in the formula.

Mistakes to Catch

1. Reading $f(x)$ as multiplication.
2. Forgetting that mathematical indexing often starts at 1.
3. Treating a set as ordered.
4. Ignoring whether interval endpoints are included.
5. Treating a tuple as unordered.
6. Memorizing symbol names without reading what the symbols do.
7. Moving past a formula before saying it in words.

Small Tests

Try these without looking back:

1. If $f(x) = x + 2$, what is $f(5)$?
2. If $A = \{2, 4, 6\}$, is $4 \in A$?
3. If $x = (8, 9, 10)$, what is $x_2$ in mathematical indexing?
4. Expand $\sum_{i=1}^{3} a_i$.
5. Say $\frac{1}{n}\sum_{i=1}^{n}\ell_i$ in words.
6. In $[0,1)$, which endpoint belongs to the interval?

If these feel unclear, return to the matching topic page. Do not rush into vectors while the notation still feels opaque.

If You Feel Lost

Return to a small example.

For a formula with a sum, use three values. For a function, plug in one number. For a set, write three members. The small case reveals the idea.

Bridge to Vectors

Vectors use this chapter immediately.

When you see

read it as:

v is an ordered pair
v_1 is the first coordinate
v_2 is the second coordinate

That is enough to begin.