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: Inconsistent systems are useless—you can prove anything from a contradiction.

Common stuck point: Inconsistency may be hidden—you don't notice until you derive a contradiction.

Sense of Study hint: Try to find a single concrete example that satisfies ALL the statements at once. If you can, they are consistent. If you cannot, look for two statements that directly conflict.

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.

Solution

  1. 1
    Examine the two conditions simultaneously: x > 5 requires x to be above 5, while x < 3 requires x to be below 3.
  2. 2
    No real number can satisfy both conditions at the same time — the requirements contradict each other.
  3. 3
    Therefore S = \emptyset. The definition is internally consistent (no logical error), but it defines the empty set.

Answer

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

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.
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Background Knowledge

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

logical statement