Analogical Reasoning Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Analogical Reasoning.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Drawing conclusions about a new situation by recognizing its structural similarity to a better-understood situation.

This is like that, so maybe what works there will work here.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Analogical reasoning draws a conclusion about a new situation from its structural likeness to a familiar one.

Common stuck point: The procedure for analogical reasoning is the easy part; the trap is trusting an analogy as proof. Asking "Am I concluding something about a new case because it maps onto a known case with the same structure?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I concluding something about a new case because it maps onto a known case with the same structure?

Worked Examples

Example 1

easy

The analogy between sets and logic:

A \cup B

(union) corresponds to

p \lor q

(OR), and

A \cap B

(intersection) corresponds to

p \land q

(AND). Use this analogy to conjecture a set version of De Morgan's law

\neg(p \lor q) \equiv \neg p \land \neg q

Answer

(A \cup B)' = A' \cap B'

First step

Map the analogy:

\neg

(negation in logic)

\leftrightarrow

complement in sets;

\lor

(OR)

\leftrightarrow

\cup

;

\land

(AND)

\leftrightarrow

\cap

Full solution

2
Translate $\neg(p \lor q) \equiv \neg p \land \neg q$ : replace each symbol — $(A \cup B)' = A' \cap B'$ .
3
Verify with a Venn diagram or specific example: let $U=\{1,2,3,4\}$ , $A=\{1,2\}$ , $B=\{2,3\}$ . $(A\cup B)'=\{4\}$ . $A'\cap B'=\{3,4\}\cap\{1,4\}=\{4\}$ . Confirmed.

Analogical reasoning transfers a known result from one domain (logic) to another (set theory) by identifying a structural correspondence. The resulting conjecture can then be verified or proved in the new domain.

Example 2

medium

Arithmetic has addition and multiplication. By analogy, what operations does set theory have, and what arithmetic laws transfer? Identify two laws that hold and one that does not.

Example 3

medium

Use analogy between real-number multiplication and matrix multiplication: what is the matrix analog of the property '

1\cdot a=a

'? Identify this matrix for

2\times 2

matrices.

Example 4

medium

Sum is to integral as product is to ___? Express the analogous operator name and write the symbol.

Example 5

medium

Integers have unique factorisation into primes. By analogy, polynomials over

\mathbb R

have unique factorisation into what?

Example 6

medium

The Pythagorean theorem

a^2+b^2=c^2

for right triangles in the plane has a 3D analog for the diagonal of a box with edges

a,b,c

. State it.

Example 7

hard

Continued analogy: sum-of-first-

n

odds equals

n^2

. By analogy with cubes, what sum gives

n^3

? State and verify for

n=3

Example 8

hard

The geometric series

\sum_{k=0}^{\infty} r^k=\frac{1}{1-r}

for

|r|<1

. By analogy, conjecture the integral formula for

\int_0^{\infty} e^{-rx}\,dx

for

r>0

Example 9

hard

Use the analogy: rate is to distance as marginal cost is to ___? Use a one-sentence justification.

Example 10

challenge

By analogy with complex numbers

a+bi

where

i^2=-1

, conjecture the structure of dual numbers used in automatic differentiation. State the defining relation and compute

(2+3\epsilon)(4+5\epsilon)

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Exponents satisfy

a^m \cdot a^n = a^{m+n}

. By analogy with functions, what does

f \circ f

suggest, and how does the notation

f^2

fit?

Example 2

medium

In arithmetic, the number

0

is the identity for addition (

a+0=a

). By analogy, identify the identity element for set union and for set intersection.

Example 3

easy

Vectors are to addition as matrices are to which operation? (Both have an additive identity and the parallel operation.) Answer with one word, the operation matrices are 'multiplied' by analogy: give 'multiplication' as

1

Example 4

easy

Square is to area as cube is to ___? A square gives a 2D measure; a cube gives which-dimensional measure? Give the dimension number.

Example 5

easy

Addition is to subtraction as multiplication is to ___? Both are inverse pairs. The inverse of multiplication is division. Compute

20\div4

Example 6

easy

Circle is to

\pi

as ___? The circle's circumference-to-diameter ratio is

\pi

. To

1

decimal, give

\pi

Example 7

easy

Velocity is to position as acceleration is to ___? Each is the rate of change of the next. Acceleration is the rate of change of velocity. If velocity rises

6

m/s in

2

s, give the acceleration.

Example 8

easy

Prime is to integers as ___ is to molecules: an indivisible building block. How many prime factors (with multiplicity) does

12

have?

Example 9

easy

Map is to territory as model is to ___? Both are simplified stand-ins. A model represents reality. If a map scale is

1:1000

, a

5

cm map distance is how many cm in reality?

Example 10

easy

Logarithm is to multiplication as ___ is to addition: an inverse-flattening tool. Since logs turn products to sums,

\log_{10}(10\times100)

equals which sum's value?

Example 11

medium

Electric current is like water flow (an analogy). Voltage is like pressure, current like flow rate. If 'flow' (current) is

2

and 'pressure' (voltage) is

6

, Ohm's analogy

V=IR

gives resistance

R=

Example 12

medium

Function composition is like a factory assembly line. If

g(x)=x+1

then

f(x)=2x

, the output of

f(g(3))

is?

Example 13

medium

A derivative is like a speedometer (instantaneous rate). For position

s(t)=t^2

, the 'speedometer' reading at

t=4

s'(4)

. Compute it.

Example 14

medium

Sets are like bags, and union is like dumping two bags together (without duplicates). For

\{1,2,3\}\cup\{3,4\}

, how many elements result?

Example 15

medium

Probability is like a slice of a pie (fraction of the whole). If an event covers

90°

of a

360°

pie, its probability is?

Example 16

medium

A system of equations is like finding where two roads cross. The lines

y=x

and

y=2

cross at what

x

Example 17

challenge

Pushing an analogy too far: the water-pipe model gives current like flow, but fails for which feature? Test the safe part: with two equal resistors in series, total resistance doubles. If each is

5

, give the total.

Example 18

challenge

Analogy is not proof: two systems behaving alike does not transfer theorems. But a valid structural analogy: complex multiplication adds angles. Multiplying numbers at angles

30°

and

40°

gives a result at what angle?

Example 19

challenge

Map-is-not-the-territory: a model approximates reality. Linear approximation of

f(x)=x^2

near

x=3

uses the tangent line

y=9+6(x-3)

. Estimate

f(3.1)

with it.

Example 20

medium

Numerator is to a fraction as dividend is to division. In

\frac{12}{4}

, the numerator

12

plays the dividend role; give the quotient.

Example 21

medium

Perimeter is to a polygon as circumference is to a circle. A square of side

5

has what perimeter?

Example 22

medium

Mean is to data as center of mass is to weights. The mean of

2,4,6

(the 'balance point') is?

Example 23

easy

By analogy with

a+0=a

for numbers, what is the additive identity for vectors?

Example 24

easy

Derivative is to slope as integral is to ___?

Example 25

easy

In arithmetic, subtraction undoes addition. In linear algebra, what undoes adding a vector

\vec v

Example 26

easy

Multiplication is to repeated addition as exponentiation is to ___?

Example 27

medium

In logic,

p\Rightarrow q

is analogous to what set relation between

\{x:p(x)\}

and

\{x:q(x)\}

Example 28

medium

By analogy with

|a-b|

measuring distance between numbers, what measures distance between two points

(x_1,y_1)

and

(x_2,y_2)

in the plane?

Example 29

medium

In arithmetic,

a\cdot 0=0

. By analogy with matrices, what is

A\cdot O

for the zero matrix

O

of compatible size?

Example 30

medium

Even/odd is to integers as ___ is to functions.

Example 31

hard

By analogy with

\frac{d}{dx}(x^n)=nx^{n-1}

, conjecture

\frac{d}{dx}(e^{kx})

Example 32

hard

By analogy with the integers, where every nonzero element has only finitely many divisors, the polynomials over

\mathbb R

have what analogous property?

Example 33

hard

Vector space is to basis as field is to ___? Give the most direct analog.

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Related Concepts

Transfer of Ideas

Background Knowledge

These ideas may be useful before you work through the harder examples.

transfer of ideas