Practice Abstraction Level in Math

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

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Quick Recap

The degree of generality at which a mathematical concept or expression is stated, ranging from specific numerical cases to fully universal symbolic forms.

$2+3=5$ is concrete. $a+b=b+a$ is abstract. 'Groups have associativity' is more abstract.

Showing a random 20 of 50 problems.

Example 1

easy

Which is more abstract: 'the area of a 3×5 rectangle is 15' or '

A = lw

Example 2

medium

Why is 'continuous function' more abstract than 'polynomial function'?

Example 3

easy

Which is the more abstract object: the number 5, or the variable

x

standing for any real number?

Example 4

medium

Which is more abstract: '

\mathbb{Z}

under addition' or 'an abelian group'?

Example 5

hard

Why is the abstract definition of 'distance' (a metric) more powerful than just

|a-b|

Example 6

challenge

Why is the theorem 'in any inner product space,

\langle x,x\rangle\ge 0

' more abstract than 'in

\mathbb{R}^n

x\cdot x\ge 0

Example 7

easy

Going from '

2,4,6,8

are even' to '

2n

is even for every integer

n

' — does abstraction increase or decrease?

Example 8

medium

True or false: a higher abstraction always makes computation easier.

Example 9

challenge

Place these on an abstraction ladder and justify the top:

5

5x

5x+3

f(x)

, 'a linear map

T

Example 10

medium

Specialize 'every continuous function on

[0,1]

attains its maximum' to a specific function.

Example 11

medium

f(x)=mx+b

, the letters

m

and

b

play a different role than

x

. Explain the abstraction-level difference.

Example 12

medium

Explain why

(a+b)^2 = a^2 + 2ab + b^2

is more useful than

5^2 = 25

Example 13

easy

Which is more abstract: '

4\cdot 7=28

' or '

ab

is a number for reals

a,b

Example 14

easy

Rank from most concrete to most abstract:

3 + 5 = 8

a + b = b + a

x + 5 = 8

Example 15

easy

Order from concrete to abstract: '7 cookies', '

n

cookies', 'a finite set'.

Example 16

easy

Three facts:

2\cdot 3=6

4\cdot 5=20

6\cdot 7=42

. Identify the abstraction step.

Example 17

medium

Generalize the operation count:

1\cdot2, 2\cdot3, 3\cdot4

are products of consecutive integers. Write the general term and state its parity.

Example 18

medium

Why does the abstract definition of a vector space let one theorem apply to arrows, polynomials, AND functions simultaneously?

Example 19

easy

Generalize:

1+2=3

2+3=5

3+4=7

. Write a general expression.

Example 20

easy

Which describes a higher abstraction level: 'the integers under addition' or 'a group'?

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Related Concepts

Variables Generalization