Binary Relations and Operations on Z

This assignment delves into mathematical reasoning by examining binary relations. It starts by proving that a specific relation defined on ordered pairs is reflexive, symmetric, and transitive. The assignment then explores defining operations like addition and multiplication on a set Z based on this relation. Finally, it defines subtraction on Z, ensuring the operands satisfy certain conditions.

Running head: MATHEMATICAL REASONING 1
Mathematical Reasoning
Name
Institution

MATHEMATICAL REASONING 2
Mathematical Reasoning
Question 1
The binary relation ~ is reflexive.
(a, b) ~ (c, d) if and only if a + d = b + c
⇒(a, b) ~ (a, b) implies that c + b = d + a
Hence a = c and b = d and since addition is commutative then the binary relation ~ is reflexive.
 The binary relation ~ is symmetric.
(a, b) ~ (c,d) implies that (b,a) ~ (c,d)
⇒ a + d = c + b implies that d + a = b + c
Since addition as a binary relation is commutative then (a, b) ~ (c,d) implies (b,a) ~ (c,d)
Thus ~ is symmetric.
 The binary relation ~ is transitive.
This means if (a, b) ~ (c, d) and (c,d) ~ (e,f) then (a,b) ~ (e,f)
⇒ a + d = c + b and c + f = e + d where f = b and a = e due to the commutativity
property of addition.
Hence c + f = e + d implies c + b = a + d which is actually (a, b) (c,d)
Thus ~is transitive.
Question 2
From the given definition [(a, b)] ~ + [(c,d)]~=[(a + c, b + d)]………………..1
⇒ if [(a,b)]~ = [(x,y)]~ then trivially a = x and b = y.
Again ⇒ if [(c,d)]~ = [(u,v)]~ then c = u and d = v.
Thus, substituting in 1. The respective values of a,b,c and d

MATHEMATICAL REASONING 3
[(x,y)]~ + [(u,v)]~ = [(x + u, y + v)]
Which is actually [(x + u, y + v)] = [(x,y)]~ + [(u,v)].
Question 3
From the definition Z = {[(a, b)] ~ such that (a, b) ∈ N x N.
Hence defining subtraction on Z [(a, b)] ~ - [(c,d)]~ = [(a – c ,b – d)] where the computations of
a – c and b – d is entirely subtraction of natural numbers and clearly
a > c and b > d.
Defining Multiplication on Z [(a, b)] ~ x [(c,d)]~ = [(a x c , b x d)] where the products
a x c and b x d are entirely computations of natural numbers.