Articles of discrete mathematics

Prove that if m and n are any two odd (integers) then mn is also odd.

Here is what i have so far By definition an integer is called odd if there exist an integer k such that n=2k+1 so if n and m are any two odd integers the product of those two integers is odd. I dont know is this is the correct way of proving that mn is […]

Cartesian products and power sets

another discrete math question. I was watching this video about Cartesian products, power sets and cardinality when a thought occurred to me. If the set S were arbitrary, would its Cartesian product be a subset of the power set? In other words would this statement be true: $S \times S \subseteq \mathcal{P}(S)$ I’ve looked around […]

How to express “exactly one” in the universe of discourse?

Lets say we have a proposition: There is exactly one car parked out side that is black. How can I express this in the universal discourse?

Find closed form for $a_{1}=2, a_{n}=a_{n-1}+n+6$

I have determined that $a_{2} = 10, a_{3} = 19, a_{4} = 29, a_{5} = 40, a_{6} = 52,$ and $a_{7} = 65$. I can see that there is a pattern in that each value increases by 8, then 9, then 10, then 11, then 12, etc. but I am having difficulty making an equation […]

Venn diagram question

Here is my question. A math examination has three questions. Twenty-six students took the examination, and every student answered at least one question. Six students did not answer the first question; twelve did not answer the second question; and five did not answer the third question. If eight students answered all three questions, how many […]

Proving if $d_0$ is the smallest positive integer in $S$ then $d_0 = \gcd(a,b)$

I would appreciate hints to this. I’ve done part (a) but am unconfident. Wondering how I could approach part (b) Question’s comment — The aim of this question is to use the Division Algorithm and the definition of greatest common divisor (gcd) to show that $d_0 = \gcd(a,b)$. Question: Let $a,b$ be integers, not both […]

Calculating the possibility of having sequential numbers in randomly picked cards

This is the question (3): I am thinking something like this: Answer will appeal when the probability of having no sequential numbers in pickings decrease below 50%. So if I assume that I picked first number as 50, then possibility of second pick to be non sequential is: $\frac{97}{99}$. 3rd is $\frac{94}{98}$, 4th is $\frac{91}{97}$and […]

Find the number of seven-letter words that use letters from the set $\{\alpha,\beta,\gamma,\delta, \epsilon\}$…

Find the number of seven-letter words that use letters from the set $\{\alpha,\beta,\gamma,\delta, \epsilon\}$ and contain at least one each of $\alpha$, $\beta$, and $\gamma$. My attempt: using inclusion/exclusion Let A denotes $\alpha$ is missing, B denotes $\beta$ is missing, and C denotes $\gamma$ is missing. Then, $$\begin{aligned}|A\cup B\cup C|&=|A|+|B|+|C|-(|AB|+|AC|+|BC|)+|ABC|\\ &= 2^7+2^7+2^7-(1+1+1)-0\\ &= 381\end{aligned}$$ $|U|= […]

If $A$, $B$, and $C$ are sets, the only way that $A\cup C = B \cup C$ is if $A=B$

If $A$, $B$, and $C$ are three sets, then the only way that $A\cup C$ can equal $B\cup C$ is $A = B$. I believe this statement is false and here is why: Let $A=\{1\}$, $B=\{2\}$, and $C=\{1,2,3,4\}$. In this scenario $A\cup C=\{1,2,3,4\}$ and $B\cup C=\{1,2,3,4\}$ however, $A\ne B$. Making the statement false. Have I […]

calculating unique value from given numbers

let’s say I have some (n) random numbers 12, 13, and 18. I want to calculate some unique value from these three such that if I change their order 13, 12, 18 or 18, 12, 13..whatever order they are in, the algorithm allows me to reproduce the same unique value from them. Also no other […]