Definitions become powerful when they are used inside proofs. Integer parity definitions and direct proofs show how “even” and “odd” are translated into algebraic forms. Once an integer is written as $2r$ or $2s+1$, addition and multiplication reveal the parity of the result. In this lesson, integer parity definitions and direct proofs are used to prove standard theorems about sums and products of even and odd integers. :::definition[Even and Odd Integers] Let $n$ be an integer. The integer $n$ is called even if there exists an integer $r$ such that $$\begin{equation*} n=2r. \end{equation*}$$ The integer $n$ is called odd if there exists an integer $s$ such that $$\begin{equation*} n=2s+1. \end{equation*}$$ ::: :::definition[Even Integer Form] For an integer $n$, $$\begin{equation*} n\ \text{is even}\leftrightarrow \exists r\in\mathbb{Z}[n=2r]. \end{equation*}$$ This form says that an even integer is twice an integer. ::: :::definition[Odd Integer Form] For an integer $n$, $$\begin{equation*} n\ \text{is odd}\leftrightarrow \exists s\in\mathbb{Z}[n=2s+1]. \end{equation*}$$ This form says that an odd integer is one more than twice an integer. ::: :::theorem[Sum of Two Odd Integers] Let $k$ and $l$ be integers. If $k$ and $l$ are both odd, then $k+l$ is even. ::: :::proof Given that $k$ and $l$ are integers. Given that $k$ and $l$ are both odd. To prove that $k+l$ is even. Since $k$ is odd, there exists an integer $a$ such that $k=2a+1$. Since $l$ is odd, there exists an integer $b$ such that $l=2b+1$. Then $$\begin{align*} k+l&=(2a+1)+(2b+1)\\ &=2a+2b+2\\ &=2(a+b+1). \end{align*}$$ Since $a+b+1$ is an integer, $k+l$ is divisible by $2$. Therefore $k+l$ is even. Hence the sum of two odd integers is even. $\square$ ::: :::definition[Sum of Two Odd Integer Forms] If $k=2a+1$ and $l=2b+1$, then $$\begin{equation*} k+l=2(a+b+1). \end{equation*}$$ This algebraic form is the reason the sum is even. ::: :::remark[Arbitrary Integers in a Universal Proof] When proving a theorem about all integers $k$ and $l$, the integers $k$ and $l$ must be chosen arbitrarily. If the proof does not depend on special values of $k$ and $l$, then universal generalization applies. ::: :::theorem[Product of Two Odd Integers] Let $k$ and $l$ be integers. If $k$ and $l$ are both odd, then $kl$ is odd. ::: :::proof Given that $k$ and $l$ are integers. Given that $k$ and $l$ are both odd. To prove that $kl$ is odd. Since $k$ is odd, there exists an integer $a$ such that $k=2a+1$. Since $l$ is odd, there exists an integer $b$ such that $l=2b+1$. Then $$\begin{align*} kl&=(2a+1)(2b+1)\\ &=4ab+2a+2b+1\\ &=2(2ab+a+b)+1. \end{align*}$$ Since $2ab+a+b$ is an integer, $kl$ has the form $2m+1$ for some integer $m$. Therefore $kl$ is odd. Hence the product of two odd integers is odd. $\square$ ::: :::definition[Product of Two Odd Integer Forms] If $k=2a+1$ and $l=2b+1$, then $$\begin{equation*} kl=2(2ab+a+b)+1. \end{equation*}$$ This algebraic form is the reason the product is odd. ::: :::theorem[Sum of Two Even Integers] Let $k$ and $l$ be integers. If $k$ and $l$ are both even, then $k+l$ is even. ::: :::proof Given that $k$ and $l$ are integers. Given that $k$ and $l$ are both even. To prove that $k+l$ is even. Since $k$ is even, there exists an integer $a$ such that $k=2a$. Since $l$ is even, there exists an integer $b$ such that $l=2b$. Then $$\begin{align*} k+l&=2a+2b\\ &=2(a+b). \end{align*}$$ Since $a+b$ is an integer, $k+l$ is divisible by $2$. Therefore $k+l$ is even. Hence the sum of two even integers is even. $\square$ ::: :::theorem[Product of Two Even Integers] Let $k$ and $l$ be integers. If $k$ and $l$ are both even, then $kl$ is even. ::: :::proof Given that $k$ and $l$ are integers. Given that $k$ and $l$ are both even. To prove that $kl$ is even. Since $k$ is even, there exists an integer $a$ such that $k=2a$. Since $l$ is even, there exists an integer $b$ such that $l=2b$. Then $$\begin{align*} kl&=(2a)(2b)\\ &=4ab\\ &=2(2ab). \end{align*}$$ Since $2ab$ is an integer, $kl$ is divisible by $2$. Therefore $kl$ is even. Hence the product of two even integers is even. $\square$ ::: A direct parity proof begins with the hypothesis and expands every parity word using its definition. If an integer is even, write it as $2r$. If an integer is odd, write it as $2s+1$. Then simplify the required expression until it has either the form $2m$ or the form $2m+1$, where $m$ is an integer. This final form proves the parity conclusion. A direct parity proof succeeds when an expression is rewritten in the defining form for even or odd integers. We observe how two chosen integers produce a sum or product and then rewrite the result as either $2m$ or $2m+1$. Enter integer values for $k$ and $l$, choose the operation, and run the checker. The output gives the parity of each input, the computed result, and the exact form that proves the result is even or odd. This helps students connect numerical experimentation with the algebraic forms used in direct proofs. :::calculator[Parity Form Checker] ::: :::exercise Prove the following using direct proof. (1) The sum of two odd integers is even. (2) The product of two odd integers is odd. (3) The sum of two even integers is even. (4) The product of two even integers is even. (5) If $k$ is even and $l$ is odd, then $k+l$ is odd. ::: :::answer (1) If $k=2a+1$ and $l=2b+1$, then $k+l=2(a+b+1)$, even. (2) If $k=2a+1$ and $l=2b+1$, then $kl=2(2ab+a+b)+1$, odd. (3) If $k=2a$ and $l=2b$, then $k+l=2(a+b)$, even. (4) If $k=2a$ and $l=2b$, then $kl=2(2ab)$, even. (5) If $k=2a$ and $l=2b+1$, then $k+l=2(a+b)+1$, odd. ::: :::faq[Frequently Asked Questions] Q: What form proves that an integer is even? A: The form $2m$, where $m$ is an integer. Q: What form proves that an integer is odd? A: The form $2m+1$, where $m$ is an integer. Q: Why must we say the new expression inside parentheses is an integer? A: The parity definition requires the multiplier or parameter to be an integer. ::: :::call-to-action[Practise Direct Proofs] subtitle: Use parity definitions carefully before studying indirect proof methods for theorem proving. button: Next Lesson | /discrete-mathematics/mathematical-logic/quantifiers-definitions-and-proofs-of-theorems/indirect-proof-methods-for-theorems button-ghost: Previous Lesson | /discrete-mathematics/mathematical-logic/quantifiers-definitions-and-proofs-of-theorems/quantifier-laws-and-existential-rules :::