False! Take b=c=2 and a=4. Then clearly a does not divide b nor c, but a divides bc.

Question

asked 2021-01-19

For each positive integer n, find the number of positive integers that are less than 210n which are odd multiples of three that are not multiples of five and are not multiples of seven. Justify your answer, which should be in terms of n.

asked 2021-02-25

For the following statement, either prove that they are true or provide a counterexample:

Let a, b, c, \(\displaystyle{m}\in{Z}\) such that m > 1. If \(\displaystyle{a}{c}\equiv{b}{c}{\left(\text{mod}{m}\right)},{t}{h}{e}{n}{a}\equiv{b}{\left(\text{mod}{m}\right)}\)

Let a, b, c, \(\displaystyle{m}\in{Z}\) such that m > 1. If \(\displaystyle{a}{c}\equiv{b}{c}{\left(\text{mod}{m}\right)},{t}{h}{e}{n}{a}\equiv{b}{\left(\text{mod}{m}\right)}\)

asked 2021-01-19

Let \(\displaystyle{F}_{{i}}\) be in the \(\displaystyle{i}^{{{t}{h}}}\) Fibonacc number, and let n be ary positive eteger \(\displaystyle\ge{3}\)

Prove that

\(\displaystyle{F}_{{n}}=\frac{1}{{4}}{\left({F}_{{{n}-{2}}}+{F}_{{n}}+{F}_{{{n}+{2}}}\right)}\)

Prove that

\(\displaystyle{F}_{{n}}=\frac{1}{{4}}{\left({F}_{{{n}-{2}}}+{F}_{{n}}+{F}_{{{n}+{2}}}\right)}\)

asked 2021-03-02

Using cardinatility of sets in discrete mathematics
the value of N is real numbers
Currently using elements of discrete mathematics by Richard Hammack chapter 18
Let A be a collection of sets such that X in A if and only if \(X \supset N\ \text{and} |X| = n\) for some n in N. Prove that \(|A| = |N|\).

asked 2021-01-30

a) If \(\displaystyle f{{\left({t}\right)}}={t}^{m}{\quad\text{and}\quad} g{{\left({t}\right)}}={t}^{n}\), where m and n are positive integers. show that
\(\displaystyle{f}\ast{g}={t}^{{{m}+{n}+{1}}}{\int_{{0}}^{{1}}}{u}^{m}{\left({1}-{u}\right)}^{n}{d}{u}\)

b) Use the convolution theorem to show that

\(\displaystyle{\int_{{0}}^{{1}}}{u}^{m}{\left({1}-{u}\right)}^{n}{d}{u}=\frac{{{m}!{n}!}}{{{\left({m}+{n}+{1}\right)}!}}\)

c) Extend the result of part b to the case where m and n are positive numbers but not necessarily integers.

b) Use the convolution theorem to show that

\(\displaystyle{\int_{{0}}^{{1}}}{u}^{m}{\left({1}-{u}\right)}^{n}{d}{u}=\frac{{{m}!{n}!}}{{{\left({m}+{n}+{1}\right)}!}}\)

c) Extend the result of part b to the case where m and n are positive numbers but not necessarily integers.

asked 2021-02-09

Suppose n is an integer. Using the definitions of even and odd, prove that n is odd if and only if 3n+1 is even.

asked 2021-01-08

Prove the following relations by Contradiction (Use rules. Don’t use truth table).

a) \(\displaystyle{\left[{B}\wedge{\left({B}\rightarrow{C}\right)}\right]}\rightarrow{C}\)

b) \(\displaystyle\neg{\left({p}\vee\neg{\left({p}∧{q}\right)}\right)}\rightarrow{q}\)

a) \(\displaystyle{\left[{B}\wedge{\left({B}\rightarrow{C}\right)}\right]}\rightarrow{C}\)

b) \(\displaystyle\neg{\left({p}\vee\neg{\left({p}∧{q}\right)}\right)}\rightarrow{q}\)

asked 2021-02-19

For the following questions you must use the rules of logic (Don’t use truth tables)

a) Show that (\(\displaystyle{p}\leftrightarrow{q}\)) and (\(\displaystyle\neg{p}\leftrightarrow\neg{q}\) ) are logically equivalent.

b) Show that not ( \(\displaystyle{p}\oplus{q}\)) and (\(\displaystyle{p}\leftrightarrow{q}\)) are logically equivalent.

a) Show that (\(\displaystyle{p}\leftrightarrow{q}\)) and (\(\displaystyle\neg{p}\leftrightarrow\neg{q}\) ) are logically equivalent.

b) Show that not ( \(\displaystyle{p}\oplus{q}\)) and (\(\displaystyle{p}\leftrightarrow{q}\)) are logically equivalent.

asked 2020-11-10

For the following, write your list in increasing order, separated by commas.

a, List the first 10 multiples of 8.

b. LIst the first 10 multiples on 12.

c. Of the lists you produced in parts a. and b., list the multiples that 8 and 12 have in common.

d. From part c., what is the smallest multiple that 8 and 12 have in common.

a, List the first 10 multiples of 8.

b. LIst the first 10 multiples on 12.

c. Of the lists you produced in parts a. and b., list the multiples that 8 and 12 have in common.

d. From part c., what is the smallest multiple that 8 and 12 have in common.

asked 2020-11-09

Use proof by Contradiction to prove that the sum of an irrational number and a rational number is irrational.