Practice Consistency (Meta) in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
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.
Example 1
easyA student claims: 'The set S = \{x \in \mathbb{R} : x > 5 \text{ and } x < 3\}.' Check whether this definition is consistent.
Example 2
mediumCheck whether the system of equations x + y = 5 and 2x + 2y = 11 is consistent.
Example 3
easyA proof assumes both 'n is even' and 'n is odd'. Is this assumption consistent? What follows?
Example 4
mediumDetermine 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.