iwonde said:(x,y)+(u,v) = (u,v)+ (x,y)
What do the equations above mean? a and b are integers, not ordered pairs.iwonde said:Homework Statement
Prove that for any two integers a and b, a+b=b+a. You may use the face that this holds for natural numbers.
Homework Equations
The Attempt at a Solution
a=(x,y), b=(u,v)
iwonde said:x,y,u,v are natural numbers
a+b = (x,y)+(u,v) = (u,v)+ (x,y) = b+a
I'm not sure if my attempt was correct.
I suspect the opening poster forgot to tell us that he's working with a specific presentation of the integers -- represented as the set of ordered pairs of natural numbers modulo an equivalence relation -- along with the definition of + that he's using.Mark44 said:What do the equations above mean? a and b are integers, not ordered pairs.
Commutativity is a property in mathematics that states that the order of operands in a mathematical operation does not affect the result. In other words, the order can be switched without changing the outcome.
To prove that a+b=b+a for integers, we can use the commutativity property of addition, which states that the sum of two numbers is the same regardless of the order in which they are added. Therefore, if we switch the order of a and b in the equation a+b=b+a, we will still get the same result.
Yes, the property of commutativity can be applied to all mathematical operations, including addition, subtraction, multiplication, and division. However, it is important to note that it may not hold true for all types of numbers, such as matrices or complex numbers.
The significance of proving a+b=b+a for integers is that it shows that the commutativity property holds true for integers. This means that the order of numbers does not matter when performing addition with integers, which can make solving equations and simplifying expressions easier.
Yes, the property of commutativity can be extended to any number of operands. This means that the order of any number of numbers can be changed without affecting the result of the operation. For example, if we have the numbers a, b, and c, the commutativity property would state that a+b+c = a+c+b = b+a+c = b+c+a = c+a+b = c+b+a.