{"id":23202,"date":"2023-03-07T06:34:20","date_gmt":"2023-03-07T06:34:20","guid":{"rendered":"https:\/\/www.booksofall.com\/pl\/?post_type=product&p=23202"},"modified":"2023-03-07T06:34:21","modified_gmt":"2023-03-07T06:34:21","slug":"introduction-to-proofs-an-inquiry-based-approach","status":"publish","type":"product","link":"https:\/\/www.booksofall.com\/pl\/introduction-to-proofs-an-inquiry-based-approach\/","title":{"rendered":"Introduction to Proofs, an Inquiry-Based approach"},"content":{"rendered":"
\n
\n

Chapter 1 – Numbers<\/h2>\n

We begin with results about the integers<\/a> Z= { . . . \u22122,\u22121,0,1,2, . . . }. In this chapter, \u201cnumber\u201d means integer. Some statements refer to the natural number<\/a>s N= {0,1,2, . . . } or the positive integers Z+ = {1,2, . . . }.<\/p>\n

Divisibility<\/h3>\n

1.1 Definition.<\/strong> For two integers d\u00a0<\/i>,n\u00a0<\/i>we say that\u00a0d divides n\u00a0<\/i>if there is an integer\u00a0k\u00a0<\/i>such that\u00a0d\u00a0<\/i>\u00b7k\u00a0<\/i>=\u00a0n<\/i>. Here,\u00a0d\u00a0<\/i>is the\u00a0divisor<\/i>,\u00a0n\u00a0<\/i>is the\u00a0dividend<\/i>, and\u00a0k\u00a0<\/i>is the\u00a0quotient<\/i>. (Alternative wordings are:\u00a0d is a factor of n<\/i>,\u00a0d goes evenly into n<\/i>,\u00a0n is divisible by d\u00a0<\/i>, or\u00a0n is a multiple of d\u00a0<\/i>.) We write\u00a0d <\/i>\u2223\u2223\u00a0n\u00a0<\/i>if\u00a0d\u00a0<\/i>is a divisor of\u00a0n<\/i>, or\u00a0d <\/i>\u2223\u2223-\u00a0n\u00a0<\/i>if it is not.<\/p>\n

1.2<\/strong> Definition<\/strong>. A number is even\u00a0<\/i>if it is divisible<\/a> by 2, otherwise it is\u00a0odd<\/i><\/a>. (We may instead say that the\u00a0parity\u00a0<\/i>is even<\/a> or odd.)
\nThe notation\u00a0d\u00a0<\/i>\u2223\u2223\u00a0n\u00a0<\/i>signifies a relationship between two integers. It is different than the fraction d<\/i>\/n<\/i>, which is a rational number: we can sensibly ask \u201cDoes 2 divide 5?\u201d but \u201cDoes 2\/5?\u201d is not sensible.<\/p>\n

1.3 Exercise<\/strong>. (Interaction with sigh) Each of these is a statement about integers. Prove each.
\nA. If a number is even then its negative is even. If a number is odd then its negative is odd.
\nB. If\u00a0d <\/i>\u2223\u2223\u00a0a\u00a0<\/i>then both \u2212d\u00a0<\/i>\u2223\u2223\u00a0a\u00a0<\/i>and\u00a0d\u00a0<\/i>\u2223\u2223\u2212a<\/i>. In addition,\u00a0d\u00a0<\/i>divides |a<\/i>| (recall that the absolute value of a number, |a<\/i>|, is\u00a0a\u00a0<\/i>if\u00a0a\u00a0<\/i>\u2265 0 and is \u2212a\u00a0<\/i>if\u00a0a\u00a0<\/i>< 0).<\/p>\n

1.4<\/strong> Exercise<\/strong>. (Interaction of party and addition) Prove or disprove. That is, for each decide if it is true or false and if it is true then prove it, while if it is false then give a counterexample.
\nA. The sum of two evens is even. The difference of two evens is even.
\nB. The sum of two odds is odd. The difference of two odds is odd.
\nC. For\u00a0a<\/i>,b\u00a0<\/i>\u2208Z, the number\u00a0a\u00a0<\/i>+b\u00a0<\/i>is even if and only if\u00a0a\u00a0<\/i>\u2212b\u00a0<\/i>is even.<\/p>\n

1.5 Exercise<\/strong>. Generalize the first item of the prior exercise to be a statement about sums of multiples of d\u00a0<\/i>\u2208Z, and then prove your statement.<\/p>\n

1.6<\/strong> Exercise<\/strong>. (Interaction of party and multiplication<\/a>) Prove or disprove. (i) The product of two evens is even. (ii) The quotient of two evens, if it is an integer, is even.<\/p>\n

1.7<\/strong> Exercise<\/strong>. For the prior exercise\u2019s first item, formulate and prove a generalization that applies to any integer.<\/p>\n

1.8<\/strong> Exercise<\/strong>. (Divisibility properties<\/a>) Prove each. Assume that all the numbers are integers.
\nA. (Reflexivity) Every number divides itself.
\nB. Every number divides 0 while the only number that 0 divides is itself.
\nC. (Transitivity) If d <\/i>\u2223\u2223\u00a0n\u00a0<\/i>and\u00a0n\u00a0<\/i>\u2223\u2223\u00a0m\u00a0<\/i>then\u00a0d <\/i>\u2223\u2223\u00a0m<\/i>. That is, if\u00a0n\u00a0<\/i>divides\u00a0m\u00a0<\/i>then so do\u00a0n<\/i>\u2019s divisors.
\nD. (Cancellation) Where d\u00a0<\/i>,n\u00a0<\/i>\u2208Z, if there is a nonzero integer\u00a0a\u00a0<\/i>such that\u00a0ad <\/i>\u2223\u2223\u00a0an\u00a0<\/i>then\u00a0d\u00a0<\/i>\u2223\u2223\u00a0n<\/i>. And, if\u00a0d\u00a0<\/i>\u2223\u2223\u00a0n\u00a0<\/i>then\u00a0ad <\/i>\u2223\u2223\u00a0an\u00a0<\/i>for all\u00a0a\u00a0<\/i>\u2208Z.
\nE. (Comparison) For d\u00a0<\/i>,n\u00a0<\/i>\u2208Z+, if\u00a0n\u00a0<\/i>is a multiple of\u00a0d\u00a0<\/i>then\u00a0n\u00a0<\/i>\u2265\u00a0d\u00a0<\/i>.
\nF. Every number is divisible by 1 and \u22121. The only numbers that divide 1 are 1 and \u22121.
\nG. The largest divisor of any nonzero\u00a0a\u00a0<\/i>\u2208Z is |a<\/i>|.
\nH. Every nonzero integer has only finitely many divisors.<\/p>\n

1.9 Exercise<\/strong>. What conclusion can you make if both a\u00a0<\/i>\u2223\u2223\u00a0b\u00a0<\/i>and\u00a0b <\/i>\u2223\u2223\u00a0a<\/i>?<\/p>\n

1.10 Exercise<\/strong>. Suppose that a<\/i>,b<\/i>,c\u00a0<\/i>\u2208Z.
\nA. Prove that if\u00a0a\u00a0<\/i>\u2223\u2223\u00a0b\u00a0<\/i>then\u00a0a <\/i>\u2223\u2223\u00a0bc\u00a0<\/i>for all integers\u00a0c<\/i>.
\nB. Prove that if\u00a0a <\/i>\u2223\u2223\u00a0b\u00a0<\/i>and\u00a0a\u00a0<\/i>\u2223\u2223\u00a0c\u00a0<\/i>then\u00a0a\u00a0<\/i>divides the sum\u00a0b\u00a0<\/i>+\u00a0c\u00a0<\/i>and difference\u00a0b\u00a0<\/i>\u2212\u00a0c<\/i>.
\nC. (Linearity) This generalizes the prior item: if a\u00a0<\/i>\u2223\u2223\u00a0b\u00a0<\/i>and\u00a0a <\/i>\u2223\u2223\u00a0c\u00a0<\/i>then\u00a0a\u00a0<\/i>divides\u00a0m\u00a0<\/i>\u00b7b<\/i>+n\u00a0<\/i>\u00b7c\u00a0<\/i>for any\u00a0m<\/i>,n\u00a0<\/i>\u2208Z.<\/p>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"