The study of discrete mathematical structures. Consider using a more specific tag instead, such as: (combinatorics), (graph-theory), (computer-science), (probability), (elementary-set-theory), (induction), (recurrence-relations), etc.

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15 views

How many different ways can $6$ chocolate bars be selected in such a way that each type is chosen at least once?

In a shop five different type of chocolates are sold. How many different ways can $6$ chocolate bars be selected in such a way that each type is chosen at least once? I know the answer is $5$. ...
-3
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1answer
12 views

Proving or disproving set statements.

I'm not sure how to approach proving or disproving these statements. I don't know where to begin, or more specifically, what it's asking me to prove or disprove. If $ A \cap B \subseteq C$ and $A ...
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7answers
53 views

Proving $\frac{n}{n+1} < \frac{n+1}{n+2}$ by induction?

I have the inequality $\frac{n}{n+1} < \frac{n+1}{n+2}$ I'm not sure how to go about proving it. I've started by testing with n = 1, which results in $\frac{1}{2} < \frac{2}{3}$ which is true ...
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1answer
26 views

How can i find equation that does not have a solution?

An operation $*$ is defined on the set $\Bbb{Z} \times \Bbb{Z}$, ie. the set containing all pairs of integers by: $$ (u,v) * (x,y)=(u+x,v \cdot y) $$ if $(\Bbb{Z} \times \Bbb{Z}, *)$ is not a group ...
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0answers
24 views

Discrete math: What is the difference between false and inverse in conditional statemensts?

Let's say there is this conditional statement: If I am in Paris, then I am in France. So, p = 'I am in Paris', and q = 'I am in France' I do not understand when p and q are false, how would that ...
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0answers
10 views

$∀ Z^+$ can be written as $c_r * 3^r + c_{r-1} * 3^{r-1} + …. + c_2 * 3^2 + c_1 * 3 + c_0$

I am trying to prove the following statement: $∀ Z^+$ can be written as $c_r * 3^r + c_{r-1} * 3^{r-1} + …. + c_2 * 3^2 + c_1 * 3 + c_0$ where $c_r =$ 1 or 2, and $c_i$ = 0, 1, or 2 for all integers ...
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1answer
27 views

Any collection of n coins can be obtained using a combination of 3¢ and 5¢ coins where n ≥ 14

I am trying to prove this statement with strong induction, but I'm a little lost on the inductive step. Proposition: Let P(n) be the sentence ‘any collection of n coins can be obtained using a ...
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2answers
24 views

Recall that $ p \rightarrow \sim q$ is equivalent to $p \land \sim q$, how can this be used as an explanation for how to use proof by contradiction.

Recall that $p \rightarrow \sim q$ is equivalent to $p \land \sim q$, how can this equivalence be used as an explanation for how to use proof by contradiction. I'm having a hard time answering this ...
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1answer
10 views

How to solve a parameteric linear equation in Zn?

Given monoid ($\Bbb{Z}$124, ⋅ ) and a parametric equation with parameter $a$ where $ax+2=5(x+a)-1$ Give the number of parameters $a$ belong $\Bbb{Z}$124 for which the above equation has precisely ...
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3answers
46 views

How to prove that $A⊆B$ means that $A∪B=B$ [duplicate]

How does one prove that $A⊆B$ means that $A∪B=B$ ? I can understand it in my head but I don't know how you'd put down in logic notation.
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1answer
13 views

convert Hex value to two's Complement

for example, let's say: 0xE5 assume the system is 8 -bit in decimal it's = 229 and in Binary it's = 1110 0101 the Two's Complement rules said: sign-bit, which's the most left, indicates a negative ...
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1answer
16 views

Rewrite the following in symbolic forms using $\sim, \land, \lor $.

Let $h = $"Peter is handsome", $c = $"Peter is clever", $o = $"Peter is optimistic". Rewrite the following in symbolic forms using $ \sim , \land, \lor $. -Peter is neither handsome, clever nor ...
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1answer
17 views

How can I show that an argument or proposition is valid through logic proof sequence?

I know the logic of proof sequence as I solved many proof problems, I now have one that has been taken my attention for a couple of days and as easy as it may look, I don't seem able to simplify the ...
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1answer
21 views

Prove that $\sum_{i=0}^{k} \lg \frac{n}{2^i} = \Theta(\lg^2 n)$

Show that if $n$ is a power of $2$, say $n = 2^k$, then we have the equality $\sum_{i=0}^{k} \lg \frac{n}{2^i} = \Theta(\lg^2 n)$. The first step is to prove $O(\lg^2n)$: $$ \lg \frac{2^k}{2^0} + \lg ...
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1answer
21 views

How can i find all invertible elements?

An operation $*$ is defined on the set $\Bbb{Z} \times \Bbb{Z}$, ie. the set containing all pairs of integers by: $$ (u,v) * (x,y)=(u+x,v \cdot y) $$ What are all the invertible elements of the ...
0
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0answers
20 views

Can you show a proof of Unique Factorization of Integers Theorem (Fundamental Theorem of Arithmetic)?

I understand the proof of "Any integer greater than 1 is divisible by a prime number" by strong mathematical induction. But I don't understand why Unique Factorization of Integers Theorem follows ...
2
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3answers
134 views

Can the cardinality of a power set ever be odd? [on hold]

Can the cardinality of a power set ever be odd? If it can, what conditions must be met?
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0answers
21 views

Integral solution of equation $Ax + By = z$ with contraints on $x, y, z$

Given $x$, $y$ and $z$, how can I check if there exists integral solution of $$Ax + By = z$$ Such that : if $x > y$, $A$ must be positive and $z \geq y$ ( Given ) if $y > x$, $B$ must be ...
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0answers
65 views

Find number of rectangles

There is $N\times M$ grid present with numbering as $1,2,\cdots,NM$ (numbering is done row wise. 1st row will contain number from $1,\cdots,M$, second row will contain $M+1,\cdots,2M$ and so on). ...
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4answers
61 views

If $a > 0$,$b>0$, and $\frac{1}{a} + \frac{1}{b}$ is an integer, prove that $a=b$. And show that $a = 1$ or $2$

If $a$ and $b$ are positive integers, and $\frac{1}{a} + \frac{1}{b}$ is an integer, prove that $a=b$. And show that $a = 1$ or $2$ -I played around with numbers and the conditions and it seems that ...
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1answer
36 views

If $m | (8n +7)$, $m | (6n + 5)$, prove that $m = ± 1$

If $m | (8n + 7)$, $m | (6n + 5)$,prove that $m = ± 1$ -We have just starting going over the "divides" notation, and I am aware of a few properties and theorems from my notes. I am; although, a bit ...
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3answers
35 views

If $\gcd(a, c) = 1$ and $b | c$, prove that $(a, b) = 1$

If $\gcd(a, c) = 1$ and $b \mid c$, prove that $(a, b) = 1$ -Not sure how to approach this problem. -We have just started the greatest common divisor section, and looking at my notes I see that ...
3
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3answers
32 views

If a is an integer, prove that $gcd(14a + 3, 21a + 4) = 1$

If a is an integer, prove that $gcd(14a + 3, 21a + 4) = 1$ -We have just started the section on greatest common divisor, one thing I know is that $gcd(a,b) = ax + by$ -My initial thought is that we ...
4
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2answers
34 views

What is the probability that these two objects are of the same color?

We have $11$ bins with $10$ objects each. Every object is either black or white, and the $i$th bin ($1 \le i \le 11$) has precisely $(i -1)$ black objects in it. Someone selects, uniformly at random, ...
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1answer
12 views

How can i show a pair forms a semigroup?

An operation . is defined on the set $Z×Z$, ie. the set containing all pairs of integers by: $(u,v).(x,y)=(u+v,v.y)$ How can i show that the pair ($Z×Z$, . ) forms a semigroup?
2
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1answer
33 views

Are these two events $A$ and $B$ independent?

Abe and Bernard are dealt five cards each from the same $52$ card deck. Let $A$ be the event that Abe gets a flush (five cards of the same suit) and $B$ be the event that Bernard’s five cards are of ...
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0answers
18 views

How can i find a semigroup?

An operation * is defined on the set $\mathbb{Z} \times \mathbb{Z}$, ie. the set containing all pairs of integers by: (u,v)*(x,y)=(u+v,v*y) How can i show that ...
0
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2answers
29 views

Find a solution $x\in\mathbb{Z_{\mathrm{784}}}$ for $x\cdot\overline{602}=\overline{308}$

I know that I have to find a positive integer $x$ that I can multiply with $602$ and then divide the result by $784$ so that the remainder of that integer division is $308$. I am sure that this is ...
2
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1answer
33 views

In how many ways can we pick a group of 3 different numbers from the group $1, 2, 3, …, 500$ such that one number is the average of the other two?

Here's the question which I'm struggling with - In how many ways can we pick a group of 3 different numbers from the group $1, 2, 3, ..., 500$ such that one number is the average of the ...
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1answer
14 views

Translate quantification into English and give the truth value

The problem is: $\exists x \in \mathbb{R} (x^3 = -1)$ I understand the following: $\exists x$ = There exists an $x$ $\in$ = shows the element before it is a member of a set after it $\mathbb{R}$ = ...
1
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1answer
20 views

equivalence relation and quotient set, Given $A = \{0,1,2,3,4,5\}$

Given $A = \{0,1,2,3,4,5\}$, Write the appropriate equivalence relation of this quotient set: $$A/_R = \{\{1,2\},\{3\},\{4,0,5\}\}$$ Well, if it was to compute $$A/_R = ...
4
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2answers
33 views

Pair of friends and a pair of “enemies” in each group of three students

The problem: There is a class. In each group of three students in the class there is a pair of friends and a pair of "enemies". Find the maximum number of students in the class. I tried to play with ...
0
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1answer
22 views

If P(i) is true for all integers i with 2≤i≤k as inductive hypothesis, then why also p(t) is true by the inductive hypothesis?

"Let P(n) be the property n is divisible by a prime number. We prove that P(n) is true for all integers n with n> 1. Basis step. If n=2, then P(n) is true because 2 is a prime and every ...
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2answers
31 views

Discrete Math: Determining if Argument is Valid

I understand there are two ways to determine validity of an argument. The first way is to construct a truth table and if the statement consisting of the premises combined together implying the ...
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0answers
15 views

Small graphs containing all trees on $n$ vertices

What do those graphs look like which contain a copy of every tree on $n$ vertices and such that no proper subgraph has this property?
0
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1answer
21 views

Determining if Argument is Valid via Short-Cut Method

I understand there are two ways to determine validity of an argument. The first way is to construct a truth table and if the statement consisting of the premises combined together implying the ...
4
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2answers
110 views

What am I counting wrong?

EDIT: I made a mistake in the beginning, the second condition has changed. Sorry for this. I'm asked to count the number of sets of 4 elements that satisfy the two following conditions: 1) Each ...
3
votes
3answers
42 views

Simplify $(k +1)! > (k + 1)^2$ to prove true for $k ≥ 4$

I am trying to prove this statement is true for $k ≥ 4$. I don't know how to work with $k + 1$ factorial, so I'm a little lost on proving this.
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2answers
16 views

Prove equivalence ratio and find class of $B\subseteq A$ , $X \sim Y \Leftrightarrow X \cap B = Y \cap B$

Prove equivalence ratio and find class of $B\subseteq A$ , $X \sim Y \Leftrightarrow X \cap B = Y \cap B$. Well, I've proved really easily that it is reflexsive, symmetrical and transitive. But I'm ...
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0answers
26 views

Number of solutions of equation with natural numbers [on hold]

Given natural numbers $s, n, k$. How to find number of solutions to equation $a_1 + a_2 + \ldots + a_s = n-s$ where $0 \leq a_i \leq k-1$ and $a_i \in \mathbb{N}$?
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3answers
53 views

Can I further simplify $5^k \cdot 5 + 9 < 6^k \cdot 6$ to prove this is true

I am trying to prove this statement, but I'm not sure where to go from here. Is don't think this is sufficiently reduced to conclude the statement is true, but I'm not positive. $k ≥ 2$ Can I ...
0
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3answers
47 views

Prove $n^3 + 7n + 3$ is divisible by 3 for all integers n ≥ 0

The statement I'm trying to prove is: $n^3 + 7n + 3$ is divisible by 3 for all integers n ≥ 0 I eventually need to prove $(k + 1)^3 + 7(k + 1) + 3$ is divisible by 3. I don't really understand ...
2
votes
1answer
45 views

Proving that a function grows faster than another

I'm told to prove or disprove that $4^{\sqrt{n}}$ grows faster than $\sqrt{4^n}$ As n tends to infinity. From my Previous years Calculus I know that if I take the derivative of two functions, and one ...
0
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1answer
39 views

Mathematical induction condition “p(k)$\Rightarrow$p(k+1)” for the divisibility by a prime number

" Mathematical induction If p(n) is a statement involving the natural number n such that: p(1) is true, and p(k)$\Rightarrow$p(k+1) for any arbitrary natural number k, then p(n) is true ...
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0answers
29 views

What is the probaility that two random permutations have same order?

I am interested in the orders of random permutations. Since the law of the log of the order of a permutation converges to a normal law (for instance Erdös-Turan Statistical group theory III), one ...
4
votes
1answer
317 views

Is there a way to find expected value of equation?

If the random variable $X$ is binomially distributed with parameters $n=6$ and $p=0.3$, what is $$E(4+3X^2)$$ I know $E(X) = np = 1.8$. I solved this problem by finding $P(X)$ of all $X$ using ...
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1answer
33 views

Is the relation $xRy$ iff $|x - y| \leq 2$ transitive?

Question: $xRy$ iff $|x-y| \leq 2$ I think I've found this to be reflexive and symmetric, but I'm stuck on transitivity. Can someone assist me with testing transitivity?
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2answers
37 views

Proof with Combinatorial Argument $\sum_{i = 1}^{n} (i-1) = nC2$

I am trying to prove below equation with combinatorial argument but I have no idea how this works. $$\sum_{i = 1}^{n} (i-1) = nC2$$ Can anyone give me a clue?
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0answers
17 views

probablitly using bayes theroem

Three numbered urns contain colored balls as described in the table below. One of the urns is picked at random and a ball is drawn from the urn; the ball is red. What is the probability the ball can ...
0
votes
1answer
14 views

Prove that: $n^2+3n^3 + 6^{lgn} is $ $\theta(n^3)$

I'm asked to prove that: $n^2+3n^3 + 6^{lgn} is $ $\theta(n^3)$ I know that for Big O, I need to show: $f(n) <= c*g(n)$ But I'm not sure how to show this, since it involves theta. Any help would ...