Practice Consistency (Meta) in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy

Are

x>5

and

x<2

consistent?

Example 2

easy

Are the statements

\{x>0,\ x<10\}

consistent?

Example 3

medium

For what

b

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

consistent?

Example 4

easy

A proof assumes both '

n

is even' and '

n

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

Example 5

hard

Determine all values of

m

for which

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

is consistent.

Example 6

medium

Are the triangle-angle conditions

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

consistent?

Example 7

medium

\{x \in \mathbb{N},\ x < 1\}

consistent (using

\mathbb{N} = \{1,2,3,\ldots\}

Example 8

medium

Adding the assumption

x=0

to the system

\{xy=1\}

: is it consistent?

Example 9

medium

Is the set of axioms

\{

'every line has

\ge2

points', 'there exists a line with exactly

1

point'

\}

consistent?

Example 10

easy

Is the system

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

consistent?

Example 11

easy

Are

\{x \text{ is even},\ x = 7\}

consistent?

Example 12

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 13

challenge

Suppose

T

is consistent. Is

T \cup \{\phi\}

guaranteed consistent for any statement

\phi

Example 14

easy

Are the constraints

x>0

and

x<5

consistent?

Example 15

medium

Determine consistency of

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

Example 16

hard

A definition states

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

. Is the question '

S \in S

?' consistent?

Example 17

easy

A figure is claimed to be both a square and a non-rectangle. Consistent?

Example 18

easy

A definition says '

0

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

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 it consistent to define

\frac{a}{0}=\infty

within standard real arithmetic?

← Back to Consistency (Meta) Worked Examples Formula Explained

Related Concepts

Logical Statement Contradiction