Practice Notation Overload in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

When the same symbol is used to mean different things in different contexts, requiring the reader to infer meaning from context.

The same word meaning different things in different conversations — context tells you which meaning applies, but this can trip up a reader who is new to the context.

Showing a random 20 of 50 problems.

Example 1

medium

The symbol

n!

uses '!' for factorial. Compute

4!

Example 2

medium

In a matrix context,

|A|

for

A = \begin{pmatrix}3 & 1\\2 & 4\end{pmatrix}

means the determinant. Compute it.

Example 3

hard

In abstract algebra,

\mathbb{Z}/n\mathbb{Z}

denotes the integers mod

n

. For

n = 5

, how many distinct elements are in this set?

Example 4

medium

The symbol

(a,b)

can mean an open interval or an ordered pair. As a gcd,

(8,12)

means

\gcd

. Compute

(8,12)

in that sense.

Example 5

easy

What does

f'(x)

mean in calculus? For

f(x)=x^2

, give

f'(3)

Example 6

challenge

In tensor calculus,

T^{ij}

T_{ij}

T^i_j

all denote different things. What standard convention disambiguates them?

Example 7

challenge

a_n^k

, what's the difference between

k

as an index and

k

as an exponent? Give an example where this distinction matters.

Example 8

easy

In the context

\{1, 2, 3\}

, what does

\{\}

mean?

Example 9

hard

f(x) = x^2

, the parentheses denote function application. In

(x+1)^2

, what do the parentheses denote? Compute when

x=3

Example 10

challenge

\binom{n}{k}

and

C(n,k)

both denote combinations, but '

nCr

' on a calculator is the same. Compute

\binom{5}{2}

Example 11

medium

The superscript in

T^{-1}

for a matrix means inverse, while

A^T

means transpose. For

A=\begin{pmatrix}1&2\\3&4\end{pmatrix}

, give the top-left entry of

A^T

Example 12

easy

The symbol

|\cdot|

is used for absolute value of numbers and cardinality of sets. Evaluate: (a)

|-3|

, (b)

|\{1,2,3\}|

Example 13

medium

\log x

can mean base-10, base-

e

, or base-2 depending on field. In CS texts, it usually means base what?

Example 14

medium

In the expression

\sin^2 x

, the superscript

2

means what? Compute

\sin^2(\pi/6)

Example 15

medium

The symbol

\ast

can mean: (1) multiplication, (2) convolution, (3) a generic binary operation. For each usage, give one example context.

Example 16

challenge

Translate carefully: the statement '

|G| = 6

and

G

is generated by elements satisfying

a^3 = e

and

b^2 = e

.' What does each symbol mean?

Example 17

medium

Set-builder

\{x \in \mathbb{R} \mid x > 0\}

: what does the

\mid

mean?

Example 18

easy

In the integral

\int_a^b f(x)\, dx

, what does

dx

mean?

Example 19

medium

Decide what

(2, 5)

means in each context: (i) coordinate plane; (ii) interval on the real line. For (ii), is

3

in the set?

Example 20

easy

\mathbb{R}^n

, what does the superscript

n

mean?

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Related Concepts

Ambiguity Symbolic Overload