Prove that if x and y are real numbers, then max (x, y)+ min(x, y)=x+y .[ Hint: Use a proof by c

step 1 Okay, so for this question we have this statement right here and we want to prove that it's true

Prove that if x and y are real numbers, then max (x, y)+...

Prove that if $x$ and $y$ are real numbers, then max $(x, y)+$ $\min (x, y)=x+y .[\text { Hint: Use a proof by cases, with the }$ two cases corresponding to $x \geq y$ and $x<y,$ respectively. $]$

Key Concepts

Proof by Cases

Proof by cases is a method used in mathematical proofs to handle different scenarios that exhaust all possibilities. In this approach, the problem is divided into distinct cases, and the statement is proven separately for each case. This method is particularly useful when the conditions of the problem change the structure or behavior of the functions involved, as seen with the maximum and minimum functions when the order of numbers switches.

Properties of Real Numbers

Properties of real numbers, such as the total order and the laws of arithmetic, provide a foundation for understanding how functions like maximum and minimum operate. These properties allow for the comparison of numbers and the application of distributive, associative, and commutative laws in proofs, ensuring that operations such as addition work consistently across different cases.

Minimum Function

The minimum function determines the smaller of two real numbers. It is defined such that for any two real numbers, the minimum is the number that is less than or equal to the other. This function is crucial for identifying the lower bound in comparisons and is directly linked with the concept of the maximum function.

Maximum Function

The maximum function identifies the larger of two real numbers. It is defined such that for any two real numbers, the maximum is the number that is greater than or equal to the other. This concept is essential for comparing values and determining which one represents the upper bound in a set of two numbers.

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