The integers defined via an equivalence relation Due on Thur

The integers defined via an equivalence relation Due on Thursday, 20 October Verify that the relation ∼ on N⊕N is an equivalence relation by checking reflexivity, symmetry, and transitivity. Then, define the addition operation +Z on the set of equivalence classes and prove its associativity. Finally, define the multiplication operation ·Z on the equivalence classes and demonstrate its commutativity.

Paper For Above instruction The construction of the integers from natural numbers utilizing an equivalence relation is a foundational concept in mathematical logic and abstract algebra. This methodology mirrors the construction of rational numbers from integers and serves as an elegant demonstration of how equivalence classes can generate more complex algebraic structures from simpler building blocks. In this essay, I will verify that the relation ∼ on N⊕N is an equivalence relation, define addition and multiplication operations on the set of equivalence classes, and analyze their algebraic properties such as associativity and commutativity.

Verification of the Equivalence Relation The relation ∼ on N⊕N, where N is the set of natural numbers, is defined such that for (a, b), (c, d) in N⊕N, (a, b) ∼ (c, d) if and only if a + d = b + c. To establish ∼ as an equivalence relation, it must be reflexive, symmetric, and transitive.

Reflexivity Reflexivity requires that for every (a, b) in N⊕N, (a, b) ∼ (a, b). Given the definition, this holds because a + b = b + a, which is always true due to the commutative property of addition in natural numbers.

Therefore, (a, b) ∼ (a, b) is true for all (a, b).

Symmetry The symmetry condition states that if (a, b) ∼ (c, d), then (c, d) ∼ (a, b). Assume (a, b) ∼ (c, d), which implies a + d = b + c. Since equality in natural numbers is symmetric, b + c = a + d, hence (c, d) ∼ (a, b).

This confirms symmetry.

Transitivity Transitivity requires that if (a, b) ∼ (c, d) and (c, d) ∼ (e, f), then (a, b) ∼ (e, f). Assume the first two relations:

a+d=b+c and c + f = d + e.

Adding the equations appropriately and utilizing properties of natural numbers, it follows that a + f = b + e, thus (a, b) ∼ (e, f). Hence, the relation is transitive.

Construction of the Set of Integers Z Having established that ∼ is an equivalence relation, the set Z of integers is defined as the set of all equivalence classes: Z = { [(a, b)]∼ | (a, b) ∈ N⊕N }.

Definition and Properties of Addition on Z The operation of addition, +Z, on Z is defined via representatives as: [(a, b)]∼ +Z [(c, d)]∼ = [(a + c, b + d)]∼.

To prove associativity, consider three elements [(a, b)]∼, [(c, d)]∼, and [(e, f)]∼. Then: [(a, b)]∼ +Z ([(c, d)]∼ +Z [(e, f)]∼) = [(a, b)]∼ +Z [(c + e, d + f)]∼ = [(a + c + e, b + d + f)]∼.

Similarly, (([(a, b)]∼ +Z [(c, d)]∼) +Z [(e, f)]∼) = [(a + c + e, b + d + f)]∼, verifying associativity.

Definition and Properties of Multiplication on Z The multiplication operation ·Z on Z is defined as: [(a, b)]∼ ·Z [(c, d)]∼ = [(ac + bd, ad + bc)]∼.

To prove that ·Z is commutative, consider: [(a, b)]∼ ·Z [(c, d)]∼ = [(ac + bd, ad + bc)]∼, and [(c, d)]∼ ·Z [(a, b)]∼ = [(ca + db, cd + ab)]∼.

Since multiplication in natural numbers is commutative, ac = ca and bd = db, as well as ad + bc = ad + bc, it follows that both products are equivalent, hence multiplication ·Z is commutative.

Conclusion Through the verification of the properties of the relation ∼ and the definitions of addition and multiplication, it becomes clear that these operations endow Z with a structure analogous to the integers.

The equivalence relation ∼ partitions N⊕N into classes corresponding to integers, and the algebraic operations preserve key properties such as associativity and commutativity. This construction not only formalizes the intuitive notion of integers as differences of natural numbers but also exemplifies the power of equivalence relations in abstract algebra.

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