Why is ordinary equality on numbers an equivalence relation?
Conditions
- Equality is viewed as a relation on numbers.
- The relation must be checked against the three properties of an equivalence relation.
Reasoning, step by step
- Check reflexivity: Verify that any number is equal to itself (e.g., ).
- Check symmetry: Verify that if a number equals another, the second equals the first (e.g., reordering still holds).
- Check transitivity: Verify that if a number equals a second, and the second equals a third, the first equals the third (e.g., chaining and yields ).
- Conclude that equality is an equivalence relation.
Example
The lecturer uses the number 3 to demonstrate the properties: "" shows reflexivity, reordering it shows symmetry, and chaining equalities shows transitivity.
Common misconceptions
- Thinking that equality is just a basic arithmetic operation and not a formal relation.
- Assuming that equality only satisfies reflexivity but fails symmetry or transitivity.
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The naive condition is insufficient because in a general commutative ring, elements can be annihilated by nonzero elements of . The refined condition requires the existence of some such that , which accounts for zero-divisor phenomena and ensures the equivalence relation is well-defined.
Conditions: is a general commutative ring (not necessarily an integral domain).; is multiplicatively closed.
No, the numerical demonstration using specific numbers like 10, 7, and 13 is merely an illustrative example, not a rigorous formal proof applicable to all integers.
Conditions: The demonstration uses specific numerical values.; A formal proof would require generalizing the property for all integers.
The reflexive property is demonstrated by showing that any number is related to itself, specifically that 10 mod 3 is equal to 10 mod 3.
Conditions: The relation is defined as equality of remainders modulo 3.; The property requires every element to be related to itself.
Transitivity is shown by demonstrating that if the first number has the same remainder modulo 3 as the second, and the second has the same remainder as the third, then the first has the same remainder as the third.
Conditions: The relation is equality of remainders modulo 3.; The property requires that if x is related to y and y is related to z, then x is related to z.
The symmetric property is demonstrated by showing that if one number has the same remainder modulo 3 as another, then the second number also has the same remainder modulo 3 as the first.
Conditions: The relation is equality of remainders modulo 3.; The property requires that if x is related to y, then y is related to x.
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