Practice Consistency (Meta) 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 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>5x>5 and x<2x<2 consistent?

Example 2

easy
Are the statements {x>0, x<10}\{x>0,\ x<10\} consistent?

Example 3

medium
For what bb is {x+y=b, x−y=2, x=3}\{x+y=b,\ x-y=2,\ x=3\} consistent?

Example 4

easy
A proof assumes both 'nn is even' and 'nn is odd'. Is this assumption consistent? What follows?

Example 5

hard
Determine all values of mm for which {y=mx, y=2x+3, x=1}\{y = mx,\ y = 2x+3,\ x = 1\} is consistent.

Example 6

medium
Are the triangle-angle conditions {∠A=70∘, ∠B=60∘, ∠C=60∘}\{\angle A=70^\circ,\ \angle B=60^\circ,\ \angle C=60^\circ\} consistent?

Example 7

medium
Is {x∈N, x<1}\{x \in \mathbb{N},\ x < 1\} consistent (using N={1,2,3,…}\mathbb{N} = \{1,2,3,\ldots\})?

Example 8

medium
Adding the assumption x=0x=0 to the system {xy=1}\{xy=1\}: is it consistent?

Example 9

medium
Is the set of axioms {\{'every line has ≥2\ge2 points', 'there exists a line with exactly 11 point'}\} consistent?

Example 10

easy
Is the system {2x=4, x=2}\{2x=4,\ x=2\} consistent?

Example 11

easy
Are {x is even, x=7}\{x \text{ is even},\ x = 7\} consistent?

Example 12

challenge
A theory has axioms A1A_1 and A2A_2. We prove A1⇒PA_1 \Rightarrow P and A2⇒¬PA_2 \Rightarrow \neg P. Is the theory consistent?

Example 13

challenge
Suppose TT is consistent. Is T∪{ϕ}T \cup \{\phi\} guaranteed consistent for any statement ϕ\phi?

Example 14

easy
Are the constraints x>0x>0 and x<5x<5 consistent?

Example 15

medium
Determine consistency of {2x+3y=12, 4x+6y=24}\{2x+3y=12,\ 4x+6y=24\}.

Example 16

hard
A definition states S={x:x∉x}S = \{x: x \notin x\}. Is the question 'S∈SS \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 '00 is both positive and negative.' Is this consistent with standard sign conventions?

Example 19

challenge
Is it consistent to have a set SS defined as 'the set of all sets that do not contain themselves'? What does this reveal?

Example 20

medium
Is it consistent to define a0=∞\frac{a}{0}=\infty within standard real arithmetic?
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