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An equivalence relation on a set is a relation that is reflexive, symmetric, and transitive.

Conditions: The relation must be defined on a set.; The relation must satisfy all three properties: reflexivity, symmetry, and transitivity.

An equivalence relation on a set is a relation that is reflexive, symmetric, and transitive.

Conditions: The relation must be defined on a set.; The relation must satisfy all three properties: reflexivity, symmetry, and transitivity.

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.

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.

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.

Ordinary equality on numbers is an equivalence relation because it satisfies the three required properties: reflexivity, symmetry, and transitivity.

Conditions: Equality is viewed as a relation on numbers.; The relation must be checked against the three properties of an equivalence relation.

Ordinary equality on numbers is an equivalence relation because it satisfies the three required properties: reflexivity, symmetry, and transitivity.

Conditions: Equality is viewed as a relation on numbers.; The relation must be checked against the three properties of an equivalence relation.

The localization $S^{-1}R$ is defined as the collection of formal fractions $r/s$ where the numerator $r$ is an element of the commutative ring $R$ and the denominator $s$ is an element of the multiplicatively closed subset $S$.

Conditions: $R$ is a commutative ring.; $S \subset R$ is a multiplicatively closed subset.; $r \in R$ and $s \in S$.

The localization $S^{-1}R$ is defined as the collection of formal fractions $r/s$ where the numerator $r$ is an element of the commutative ring $R$ and the denominator $s$ is an element of the multiplicatively closed subset $S$.

Conditions: $R$ is a commutative ring.; $S \subset R$ is a multiplicatively closed subset.; $r \in R$ and $s \in S$.