In the set of animals, a relation R is defined in each question. a R b if ^'' a and b are in di

step 1 In the question, it is given that animal A is related to animal B if and only if A comma B belon

In the set of animals, a relation R is defined in each...

In the set of animals, a relation $R$ is defined in each question.
a $\mathrm{R} \mathrm{b}$ if ${ }^{\prime \prime} \mathrm{a}$ and $\mathrm{b}$ are in different zoological parks," then $\mathrm{R}$ is
(1) only reflexive
(2) only symmetric
(3) only transitive
(4) equivalence

Key Concepts

Transitivity

Transitivity is a property where if an element a is related to an element b and b is related to an element c, then a must also be related to c. For a relation on a set, ensuring this property guarantees a consistent chain of relatedness among the elements, which is fundamental for certain types of relations such as orderings or equivalence relations.

Symmetry

Symmetry refers to a property of a relation where if an element a is related to an element b, then b is also related to a. That is, for any elements a and b in a set, if a R b holds, then b R a must also hold. This property is crucial for understanding bidirectional relationships between elements.

Equivalence Relation

An equivalence relation is a relation that satisfies three key properties: reflexivity, symmetry, and transitivity. Such relations partition a set into equivalence classes where every element is related to itself and any two elements in the same class are related under the defined relation. Equivalence relations are foundational in the study of partitions and classifications within sets.

Relation

A relation is a set of ordered pairs that establishes a connection between elements of a set. In abstract algebra and discrete mathematics, relations are used to formalize comparisons or associations among elements of a set, serving as the basis for exploring more structured relationships such as orderings or equivalences.

Reflexivity

Reflexivity is a property of a relation where every element is related to itself. In other words, for a relation R on a set, every element a must satisfy a R a. This property is examined to determine if self-relatedness is inherent in the definition of the relation.

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