Consistency (Meta) Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
A student claims: 'The set S={xāˆˆR:x>5Ā andĀ x<3}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>5x > 5 requires xx to be above 5, while x<3x < 3 requires xx to be below 3.
  2. 2
    No real number can satisfy both conditions at the same time ā€” the requirements contradict each other.
  3. 3
    Therefore S=āˆ…S = \emptyset. The definition is internally consistent (no logical error), but it defines the empty set.

Answer

S=āˆ…Ā (theĀ conditionsĀ areĀ contradictory;Ā theĀ setĀ isĀ empty)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 ā€” but the set may still be empty. Here the conditions are mutually exclusive, yielding the empty set.

About Consistency (Meta)

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

Learn more about Consistency (Meta) ā†’

More Consistency (Meta) Examples

Example 2 medium

Check whether the system of equations [formula] and [formula] is consistent.

Example 3 easy

A proof assumes both '[formula] is even' and '[formula] is odd'. Is this assumption consistent? What

Example 4 medium

Determine whether the conditions '[formula] is a prime number' and '[formula] is divisible by 4' are

ā† All Consistency (Meta) Examples Consistency (Meta) Concept