DISCRETE MATH 2030 - QUIZ 3 REVIEW

Boxes, each capable of holding 36 units, are used to ship a product from the manufacturer to a wholesaler. Express the number of boxes that would be required to ship n units of the product using either the floor or the ceiling notation. Which notation is more appropriate?

Ceiling would be more appropriate, because you need to use a quarter of a box to ship the remainder of the product. [N/36]

Theorem 4.3.1 - Every Integer is a _______________

Theorem 4.3.1 - Every Integer is a _______________ Rational number

Definition of Divisibility

If n and d are integers then: n is divisible by d if, and only if, n equals d times some integer and d ≠ 0. Instead of "n is divisible by d," we can say that: n is a multiple of d, or d is a factor of n , or d is a divisor of n, or d divides n. The notation d|n is read "d divides n." Symbolically, if n and d are integers: d|n ⟺ ∃ an integer, say k, such that n = dk and d ≠ 0. The notation d ⫮ n is read "d does not divide n."

If k is an integer, what is [k + 1/2]?

K + 1 is the ceiling.

Corollary

The double of a rational number is rational. This is a statement whose truth can be immediately deduced from a theorem that has already been proved. Proof Suppose r is any rational number. Then 2r = r + r+ r is a sum of two rational numbers. So, by Theorem 4.3.2, 2r is rational.

Suppose a, b, c, and d are integers and a ≠. Suppose also that x is a real number that satisfies the equation ax + b / cx + d = 1 Must x be rational? If so, express x as a ratio of two integers.

We want to prove if x is rational. So: Let me try to see what the number really is by solving for x. But we don't know what x is. So we may try to solve for x. Multiply the denominator to get rid of it. Multiply top and bottom by cx + d. Now you get this: The equation is the problem is equivalent to ax + b = cx + d whenever cx + d does not = 0 Then solve for x cx goes to the left and b goes to the right = ax - cx = d - b Then x (a - c) = d - b Since a does not equal c Then x = a - b/ d - c So by the definition of the proof, Since d - b, a - c are integers and a - c does not = 0, the we see that x must be rational. ***Always remember to use English like above to express proofs. Analogy - Math is a fish and English is water Fish can't live without water

Rational

A real number r is rational if, and only if, it can be expressed as a quotient of two integers with a nonzero denominator. A real number that is not rational is irrational. More formally, if r is a real number, then R is rational ↔ ∃ integers a and b such that r = a/b and b ≠0

Zero Product Property

If neither of two real numbers is zero, then their product is also not zero. (It follows that (m + n)/mn is a quotient of two integers with a nonzero denominator and hence is a rational number.

If k is an integer, what is [k]?

K is the ceiling.

Helpful hints

LOOK AT Q AS THE QUOTIENT AND R AS THE REMAINDER When dealing with fractions, d is the denominator, and n is the numerator. You can never have a zero value in the denominator.

derive the statements as corollaries of Theorems 4.3.1, 4.3.2. *Theorem 4.3.1 = Every integer is a rational number. **Theorem 4.3.2 = The sum of any two rational numbers is rational. If r is any rational number, then 3r2 - 2r + 4 is rational

Since r is rational, then r squared is rational, and that multiplied times 3 is rational, and 2 times a rational number is rational, and the product of all the numbers is rational, then r is rational.

Properties of even and odd integers

Properties of even and odd integers 1. The sum, product, and difference of any two even integers are even. 2. The sum and difference of any two odd integers are even. 3. The product of any two odd integers is odd. 4. The product of any even integer and any odd integer is even. 5. The sum of any odd integer and any even integer is odd. 6. The difference of any odd integer minus any even integer is odd. 7. The difference of any even integer minus any odd integer is odd.

Number Theory

The study of properties of integers

What does the symbol " | " stand for

This symbol stands for the word "divides" which will give you an integer. **Note: The " / " symbol does not always give you a result of an integer, so be careful. In other words, a | b denotes the sentence " a divides b," where as a / b denotes the number a divided by b.

Give a reason for your answer. Assume that all variables represent integers. If n = 4k + 3, does 8 divide n2 - 1?

Yes, Let k be an integer. Then n = 4k + 3 is also an integer And n2 - 1 = (4k + 3)2 -1 16k2 + 24k + 9 - 1 16k2 + 24k + 8 8 (2k2 + 3k + 1) Since 2k2 + 3k + 1 is an integer, we see that n 2 - 1 is an integer divisible by 8.

Divisibility Problem Is 7 a factor of -7?

Yes, -7 = 7 * (-1)

Divisibility Problem Is 21 divisible by 3?

Yes, 21 = 3 * 7.

Divisibility Problem Is 32 a multiple of -16?

Yes, 32 = (-16) * (-2).

Give a reason for your answer. Assume that all variables represent integers. Is 4 a factor of 2a * 34b?

Yes, 4 * ( a * 17) = 2a *2 * 17b

Divisibility Problem Does 5 divide 40?

Yes, 40 = 5 * 8.

Divisibility Problem Does 7 | 42?

Yes, 42 = 7 * 6.

Divisibility Problem Is 6 a factor of 54?

Yes, 54 = 6 * 9.