b We can use calculus to show that the minimum value of ݔ � 2ݔ2 is 1 Let ሺݔሻݔ ଶ

b) We can use calculus to show that the minimum value of ݔ ଶ +2ݔ+2 is 1 . Let ݂ሺݔሻ=ݔ ଶ +2ݔ+2 Then ݂ ᇱ ሺݔሻ=2ݔ+2 and ݂ ᇱ ሺݔሻ=0 when ݔ =−1 . Since ݂ ᇱᇱ ሺݔሻ=2 , which is positive, the second derivative test tells is that ݔ =−1 is a minimum. The minimum value of ݂ is ݂ሺ−1ሻ=ሺ−1ሻ ଶ +2ሺ−1ሻ+2=1 , so ݂ሺݔሻ≥1 and ݂ሺݔሻ≠0 for any real number ݔ . c) If we plot the curve ݕ =ݔ ଶ +2ݔ+2 , we find that the curve does not intersect the ݔ -axis, signifying that ݕ ≠0 for any ݔ . This example shows us that in the actual carrying out a mathematical proof, you need “mathematical knowledge” in addition to knowing the structure or what it means to give a formal, rigorous proof. In this course, we focus on proof techniques, but we also do not shy away from alluding to or including relevant “mathematical knowledge” to enrich our study.

12 4. Definitions, Theorems, and Proofs We have introduced the basic elements of mathematical statements and arguments, and have seen some examples of proofs. In this course we also hope to learn to appreciate the fact that there are many ways of writing a correct proof, and of course more ways of writing a wrong proof! Even among correct proofs of the same theorem, you may “like” some proofs more than the others. In any case, our goal is two-fold: to write a proof that is clear and “easy” to follow and to understand a well-written proof. In this section, we will present examples of writing a proof in different ways. First let us gather all the definitions we may use. Definition 1 An integer ݔ is even if ݔ =2݊ for some integer ݊ . Definition 2 An integer ݔ is odd if ݔ =2݊+1 for some integer ݊ . Definition 3 An integer ܽ is divisible by an integer ܾ (denoted ܾ|ܽ ), if ܾܿ =ܽ for some integer ܿ . We also say “ ܾ divides ܽ ”, “ ܾ is a factor of ܽ ”, “ ܾ is a divisor of ܽ ”, or “ ܽ is a multiple of ܾ ”. Definition 4 An integer ݌ is prime if ݌ >1 and the only positive divisors of ݌ are 1 and ݌ . Definition 5 An integer is composite if there is an integer ܾ such that ܾ|ܽ and 1<ܾ <ܽ . Example 1 Prove the following proposition. Proposition An integer ݔ is odd if and only if ݔ+1 is even. Proof 1 The proposition is a biconditional, and we need to prove in two directions. ( ⇒ ) We show that if ݔ is odd then ݔ+1 is even: ݔ is odd → ݔ =2݊+1 for some integer ݊ (Definition 2) → ݔ+1=2݊+1+1=2݊+2=2ሺ݊+1ሻ → ݔ+1=2݇ , where ݇ =݊+1 is an integer → ݔ+1 is even (Definition 1) ( ⇐ ) We show that if ݔ+1 is even, then ݔ is odd: ݔ+1 is even → ݔ+1=2݊ for some integer ݊ (Definition 1) → ݔ =2݊−1=2ሺ݊−1ሻ+1 , where ݊−1 is an integer → ݔ =2݇+1 , where ݇ =݊−1 is an integer → ݔ is odd (Definition 2) ■ Remarks: • The symbol ■ signifies the end of a proof.