Consistency (Meta) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Consistency (Meta).

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

Concept Recap

The property of a set of mathematical statements having no internal contradictions — all statements can be simultaneously true within the same system.

Imagine building with a set of rules: if one rule says 'the door must be open' and another says 'the door must be closed,' the system is inconsistent and no valid state exists. Consistency matters because from a single contradiction you can logically derive any statement at all (the principle of explosion), making the entire system meaningless.

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: A set of statements is consistent if they can all be true at once — no contradiction can be derived from them.

Common stuck point: The procedure for consistency (meta) is the easy part; the trap is confusing consistency with completeness. Asking "Can every statement in this set be true at the same time without forcing a contradiction?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can every statement in this set be true at the same time without forcing a contradiction?

Worked Examples

Example 1

easy

A student claims: 'The set

S = \{x \in \mathbb{R} : x > 5 \text{ and } x < 3\}

.' Check whether this definition is consistent.

Answer

S = \emptyset \text{ (the conditions are contradictory; the set is empty)}

First step

Examine the two conditions simultaneously:

x > 5

requires

x

to be above 5, while

x < 3

requires

x

to be below 3.

Full solution

2
No real number can satisfy both conditions at the same time — the requirements contradict each other.
3
Therefore $S = \emptyset$ . The definition is internally consistent (no logical error), but it defines the empty set.

A set definition is consistent if no logical contradiction prevents elements from being described — but the set may still be empty. Here the conditions are mutually exclusive, yielding the empty set.

Example 2

medium

Check whether the system of equations

x + y = 5

and

2x + 2y = 11

is consistent.

Example 3

hard

Show that adding '

0 = 1

' to ordinary arithmetic makes it inconsistent.

Practice Problems

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

Example 1

easy

A proof assumes both '

n

is even' and '

n

is odd'. Is this assumption consistent? What follows?

Example 2

medium

Determine whether the conditions '

n

is a prime number' and '

n

is divisible by 4' are consistent. If so, find an example; if not, explain why.

Example 3

easy

Is the system

\{x+y=5,\ x+y=3\}

consistent?

Example 4

easy

\{x=2,\ y=3,\ x+y=5\}

consistent?

Example 5

easy

Does the single equation

x^2=-1

have a real solution? Is it consistent over the reals?

Example 6

easy

Are the constraints

x>0

and

x<5

consistent?

Example 7

easy

Are

x>5

and

x<2

consistent?

Example 8

easy

A definition says '

0

is both positive and negative.' Is this consistent with standard sign conventions?

Example 9

easy

Is the system

\{2x=4,\ x=2\}

consistent?

Example 10

easy

Can a triangle have side lengths

1, 1, 5

? Is this consistent with the triangle inequality?

Example 11

medium

For what value of

k

is the system

\{x+y=2,\ 2x+2y=k\}

consistent?

Example 12

medium

Is it consistent to assume both '

n

is the largest integer' and the rules of arithmetic?

Example 13

medium

Are the three statements

a=b

b=c

a\ne c

consistent?

Example 14

medium

Is a consistent set of statements necessarily all true? Decide with

\{

'unicorns have one horn', 'no unicorns exist'

\}

Example 15

medium

Adding the assumption

x=0

to the system

\{xy=1\}

: is it consistent?

Example 16

medium

Is the set of axioms

\{

'every line has

\ge2

points', 'there exists a line with exactly

1

point'

\}

consistent?

Example 17

challenge

Determine all

k

for which

\{x+y+z=1,\ x+y=k,\ z=2\}

is consistent.

Example 18

challenge

A student claims

0.999\ldots \ne 1

while accepting standard real-number axioms. Is this position consistent? Explain.

Example 19

challenge

Is it consistent to have a set

S

defined as 'the set of all sets that do not contain themselves'? What does this reveal?

Example 20

medium

Is the pair of constraints

x+y=10

and

x-y=2

consistent? If so, solve.

Example 21

medium

Is it consistent to define

\frac{a}{0}=\infty

within standard real arithmetic?

Example 22

medium

Are

\sin\theta=\frac12

and

\cos\theta=\frac12

consistent for a single real

\theta

Example 23

easy

Is the set

\{x \in \mathbb{Z}: x>3 \text{ and } x<8\}

defined by consistent conditions?

Example 24

easy

Are

\{a+b=10,\ a=4\}

consistent?

Example 25

medium

For which

k

\{x+y=3,\ 2x+2y=k\}

consistent?

Example 26

medium

\{x^2 = 4,\ x > 0\}

consistent? If so, give the solution.

Example 27

medium

\{x^2 = -4,\ x \in \mathbb{R}\}

consistent?

Example 28

medium

\{n \text{ is prime},\ n \text{ is even},\ n \ne 2\}

consistent?

Example 29

medium

Are the triangle-angle conditions

\{\angle A=70^\circ,\ \angle B=60^\circ,\ \angle C=60^\circ\}

consistent?

Example 30

medium

For what

b

\{x+y=b,\ x-y=2,\ x=3\}

consistent?

Example 31

medium

Are the conditions

\{f(x)=x^2,\ f(3)=10\}

consistent?

Example 32

medium

Determine consistency of

\{2x+3y=12,\ 4x+6y=24\}

Example 33

hard

Find all

a

for which

\{x^2+y^2=a,\ x+y=2,\ x=y\}

is consistent.

Example 34

hard

Are the statements

\{

P

is true', '

\neg P

is true'

\}

consistent?

Example 35

hard

A definition states

S = \{x: x \notin x\}

. Is the question '

S \in S

?' consistent?

Example 36

hard

For what real

c

\{\sin\theta = c,\ \theta \in \mathbb{R}\}

consistent?

Example 37

hard

Determine all values of

m

for which

\{y = mx,\ y = 2x+3,\ x = 1\}

is consistent.

Example 38

challenge

A theory has axioms

A_1

and

A_2

. We prove

A_1 \Rightarrow P

and

A_2 \Rightarrow \neg P

. Is the theory consistent?

Example 39

challenge

Consider non-Euclidean geometry that drops the parallel postulate. Is the result consistent?

Example 40

challenge

Suppose

T

is consistent. Is

T \cup \{\phi\}

guaranteed consistent for any statement

\phi

← Back to Consistency (Meta) Formula Explained Practice Mode

Related Concepts

Logical Statement Contradiction

Background Knowledge

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

logical statement