Let σ := (R), where R is a binary relation symbol. Write down a σ-theory T whose models are exactly the equivalence relations with infinitely many classes.  More precisely, for each σ-structure M := (M, RM), M ⊨ T if and only if RM is an equivalence relation on M with infinitely many equivalence classes. No proof needed but please explain the answer.

In Exercises 610, a relation R is defined on the set Z of all integers. In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and list at least four members of each. xRy if and only if x+3y is a multiple of 4.

Label each of the following statements as either true or false. If R is an equivalence relation on a nonempty set A, then any two equivalence classes of R contain the same number of element.

True or False Label each of the following statements as either true or false. Let be an equivalence relation on a nonempty setand let and be in. If, then.

In Exercises , a relation is defined on the set of all integers. In each case, prove that is an equivalence relation. Find the distinct equivalence classes of and list at least four members of each. 10. if and only if .

Determine whether the set S={1,x2,2+x2} spans P2.

Let R be the relation defined on the set of integers by aRb if and only if ab. Prove or disprove that R is an equivalence relation.

In each of the following parts, a relation is defined on the set of all human beings. Determine whether the relation is reflective, symmetric, or transitive. Justify your answers. xRy if and only if x lives within 400 miles of y. xRy if and only if x is the father of y. xRy if and only if x is a first cousin of y. xRy if and only if x and y were born in the same year. xRy if and only if x and y have the same mother. xRy if and only if x and y have the same hair colour.

In Exercises 610, a relation R is defined on the set Z of all integers, In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and least four members of each. xRy if and only if x2+y2 is a multiple of 2.