Tagged Questions
0
votes
3answers
49 views
Pigeon holes principle
Let $P$ be a group that it's elements are 257 sentences in which only atomic sentences from $A,B,C$ exist (i.e. $A \iff B,\space\space A \wedge B \wedge C, \space\space...$) Show that there exists two ...
3
votes
4answers
67 views
Transitivity on Relations
I have a question concerning proving properties of Relations. The question is this: How would I go about proving that, if R and S (R and S both being different Relations) are transitive, then R union ...
1
vote
4answers
74 views
Writing an expression using logic
Write an expression using letters $\land, \lor, and$ $\neg$ which has the following truth table:
$$\begin{array}{ccc|c}
P&Q&R&???\\ \hline
T&T&T&F\\
T&T&F&T\\
...
2
votes
2answers
58 views
Are these propositions equivalent?
Statement 1: Maria will find job if she learns mathematics.
Statement 2: Maria will find a job unless she does not learn
mathematics.
I know the answer is probably that these are same, but ...
9
votes
2answers
377 views
9 pirates have to divide 1000 coins…
A band of 9 pirates have just finished their latest conquest - looting, killing and sinking a ship. The loot amounts to 1000 gold coins.
Arriving on a deserted island, they now have to split up the ...
1
vote
3answers
55 views
Prove that Statements forms are tautologies
Given variable statement forms $A$ and $B$. How to prove that if $(A\land B)$ is a tautology then $A$ and $B$ are tautologies too?.
Mi approach would be a proof by contradiction, something like: If ...
1
vote
2answers
52 views
Probe logical equivalence
a. $\quad\quad p \rightarrow q\;\equiv\;-p ∨ q$
b. $\quad -(p \land q)\;\equiv\; -p \lor -q$
Can these be proven without truth tables?
1
vote
2answers
83 views
Prove a tautology using truth table
How do I prove $(\lnot p \rightarrow F)\rightarrow (p=T)\;$ using a truth table?
(This tautology symbolizes a "proof by contradiction". If p being false leads to a contradiction, then p is true.)
3
votes
2answers
77 views
$\forall x \in I , \exists y \in I$ such that $xy \in I $
I just have a small question! Really basic I'm sure but something is bothering me. Take note of the following statement:
$\forall x \in I , \exists y \in I$ such that $xy \in I $
Does this ...
3
votes
4answers
126 views
If $b\mid ca$, then $b\mid a$. Is this true?
My proof: We want to show $b\mid a$ i.e. $a = bn$ for some integer $n$. Since $b\mid ca$, $ca = bm$ for some integer $m$. Substituting for $a$ gives us $c(bn) = bm \Rightarrow b(cn) = bm\dots$
After ...
0
votes
1answer
60 views
Proving that $R$ is an equivalence relation.
Let $A$ be the set of all statement forms in three variables $p$, $q$, and $r$. Let $R$ be the relation defined on $A$ as follows: For all $P$ and $Q$ in $A$,
$$P\; R\; Q \longleftrightarrow P\; ...
2
votes
2answers
76 views
Propositional Calculus Questions
I have a few questions that I am working on, that I supposedly answered incorrectly.
I have the following statements that I am charged to express in symbolic form:
$f =$ you are a full-time student; ...
1
vote
5answers
109 views
Definition Of Symmetric Difference
The definition of a symmetric difference of two sets, that my book provides, is: Set containing those elements in either $A$or $B$, but not in both $A$ and $B$.
So, in set builder notation, I figured ...
1
vote
2answers
72 views
Help with sets and subsets
Show that:
if $A \subseteq C\,$ and $\,B \subseteq D,\,$ then $\,A \times B \subseteq C \times D.$
Can anyone help me with this?
2
votes
1answer
141 views
Translating a sentence into symbolic form.
If G(x) = "x is green" and the sentence is "Some animals are green and some are not green."
Then is my symbolic sentence correct? $$\exists x G(x) \land \exists y \lnot G(y)$$
1
vote
1answer
86 views
The truth value of quantified statements
I just took an exam and the following problems were asked:
Determine the truth value of each of these statements if the domain
consists of all real numbers.
$\forall x \forall y \; ...
3
votes
3answers
127 views
Proving DeMorgan's Theorem
I'm trying to prove that (without using logical equivalencies):
$\overline{A\cap B} = \bar A \cup \bar B$
by proving both sides:
(1) $ x \in \overline{A\cap B} \to x \in \bar A\cup\bar B$
(2) $ x ...
0
votes
2answers
43 views
Trouble with proving whether argument is valid or not
I nee to determine whether these arguments are valid or not... How can i go abouts solving this question? I am having trouble finding a theoretical way to prove this... There are four of them, perhaps ...
0
votes
3answers
35 views
Identifying Proof Method and Implementing It
The question I am working on is:
Prove that if $m+n$ and $n+p$ are even integers, where
$m$, $n$,and $p$ are integers, then $m+p$ is even. What kind
of proof did you use?
I was thinking--and ...
1
vote
2answers
106 views
Negating Quantified statements
The problem I am working on is:
Express each of these statements using quantifiers. Then
form the negation of the statement, so that no negation
is to the left of a quantifier. Next, express ...
2
votes
1answer
101 views
Determining The Truth Value Of Quantified Statements
The problem I am working on is:
Determine the truth value of each of these statements if
the domain consists of all integers.
a) $∀n(n+1>n)$
b) $∃n(2n=3n)$
c) $∃n(n=−n)$
...
3
votes
3answers
198 views
Determining Whether Arguments Are Valid
The question is, "Determine whether each of these arguments is valid. If an
argument is correct, what rule of inference is being used? If it is not, what logical error occurs?
a) If $n$ is a real ...
1
vote
4answers
127 views
Proving that for any sets $A,B,C$, and $D$, if $(A\times B)\cap (C\times D)=\emptyset $, then $A \cap C = \emptyset $ or $B \cap D = \emptyset $
I'm trying to prove that for any sets $A$, $B$, $C$, and $D$, if the Cartesian product of $A$ and $B$ is disjoint with the Cartesian product of $C$ and $D$, then either $A$ and $C$ are disjoint or $B$ ...
2
votes
0answers
108 views
What exactly is written on this blackboard? [closed]
The other day I walked into an office in the university where they are working on some kind of a project which I don't really know a lot about, and I found the blackboard filled with the following ...
3
votes
1answer
145 views
How to use the Rules of Inference to a statement from two premises
The problem is as follows: Given the premise ∀x(P (x) ∨ Q(x)) and ∀x((¬P (x) ∧ Q(x)) → R(x)) is true, use the rules of inference to show that ∀x(¬R(x) → P(x)) is also true. (The domains of all ...
2
votes
4answers
79 views
Quantified Statements To English
The problem I am working on is:
Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people.
a) $∀x(C(x)→F(x))$
...
9
votes
2answers
146 views
What is the converse of this statement and is it true?
If $a$ and $b$ are relatively prime, $a\mid c$ and $b\mid c$, then $(ab)\mid c$.
I am lost. Would the converse be "If $(ab)\mid c$, then $a$ and $b$ are relatively prime and $a\mid c$ and $b\mid c$" ...
3
votes
4answers
146 views
Quantifies, predicates, logical equivalence
I am asked if $(\exists x) (P(x) \rightarrow Q(x))$ and $\forall x P(x) \rightarrow \exists xQ(x)$ are logically equivalent. I dont think they are but how will I prove it. Am I supposed to use either ...
2
votes
2answers
165 views
Converting $\exists x \exists y (x\geq y)$ into English
$\exists x \exists y (x\geq y)$
The universe of discourse is all real numbers. This says that there exists an $x$ and there exists a $y$ such that $x\geq y$. But what is this actually trying to say? ...
1
vote
1answer
138 views
Sodoku Puzzles and Propositional Logic
I am currently reading about how to solve Sudoku puzzles using propositional logic. More specific, they use the compound statement $\bigwedge_{i=1}^{9} \bigwedge_{n=1}^{9} \bigvee_{j=1}^{9}~p(i,j,n)$, ...
2
votes
1answer
99 views
Are these statements logically equivalent? (quantifiers)
Is $\forall x(P(x) \vee Q(y))$ the same as $(\forall x P(x)) \vee Q(y)$?
I understand that if I had $\forall x(P(x) \vee Q(x))$, that it is not the same as $(\forall x P(x)) \vee (\forall x Q(x))$. ...
0
votes
1answer
128 views
Proof of logical equivalence of biconditional and other proposition
I am working on a problem where I need to show the logical equivalence of two propositions. One is a biconditional: p<->q. And the other is this: $(p \land q) \land (\lnot p \land \lnot q)$
I ...
3
votes
5answers
93 views
Quantification over the empty set [duplicate]
Possible Duplicate:
Why is predicate “all” as in all(SET) true if the SET is empty?
In don't quite understand this quantification over the empty set:
$\forall y \in \emptyset: Q(y)$
The ...
3
votes
2answers
184 views
Rule of inference for proof by contradiction.
In the book "Discrete Mathematical Structures" - Kolman, author has stated that proof by contradiction is based on the tautology ((p⇒q)∧(~q))⇒(~p).And that this argument form is often applied to the ...
12
votes
5answers
392 views
Notation Question: What does $\vdash$ mean in logic?
In a "math structures" class at the community college I'm attending (uses the book Discrete Math by Epp, and is basically a discrete math "light" edition), we've been covering some basic logic.
I've ...
6
votes
2answers
239 views
Visualizing Concepts in Mathematical Logic
If you were forced to speculate or offer anecdotal evidence, how would you say excellent practicioners of mathematical logic coneptually grasp statements like:
$$ \vdash ((P \rightarrow Q) ...
2
votes
2answers
77 views
Express each of these sentences in terms of $Q(x, y)$, quantifiers, and logical connectives,
Let Q(x,y) be the statement “x has been a contestant on quiz show y”, where the
domain of x is the set of students and the domain for y consists of all
quiz shows. For each of the English sentences ...
5
votes
1answer
144 views
What are a list of helpful boolean identities for solving boolean functions?
For instance, things like $P \Leftrightarrow Q \equiv (P \Rightarrow Q) \land (Q \Rightarrow P)$ is a very helpful formula to know, as is $P \Rightarrow Q \equiv \lnot P \lor Q$ is another helpful ...
4
votes
3answers
96 views
How does this textbook go from this step to the next in proving this?
Here's the picture of the question:
How does it go from p v ~q to ~p -> ~q?
2
votes
2answers
111 views
Rewriting Conditionals In Their Well Known Form
The question is,
"Write each of these statements in the form “if p, then q” in English. [Hint:Refer to the list of common ways to express conditional statements.]
a) It snows whenever the wind ...
2
votes
3answers
109 views
Writing Propositions With Propositional Variables
The puzzle I am working on is:
"Let $p$, $q$, and $r$ be the propositions
$p$: Grizzly bears have been seen in the area.
$q$: Hiking is safe on the trail.
$r$: Berries are ripe along the trail.
...
2
votes
1answer
57 views
Transcribing Propositions In English To The Language Of Logic
The question I am working on is:
Let $p$ and$q$ be the propositions
$p$:It is below freezing.
$q$:It is snowing.
Write these propositions using $p$ and $q$ and logical connectives (including ...
1
vote
1answer
49 views
Determine Consistency Of System Specifications
I am looking at an example problem in my text:
"Determine whether these system specifications are consistent:
'The diagnostic message is stored in the buffer or it is re-transmitted.'
'The ...
1
vote
1answer
239 views
Strong induction proofs
I'm having trouble understanding strong induction proofs
I understand how to do ordinary induction proofs and I understand that strong induction proofs are the same as ordinary with the exception ...
1
vote
1answer
232 views
Necessary and sufficient conditions?
a. if p IMPLIES q, then p is a SUFFICIENT condition for q
b. if NOT p IMPLIES NOT q, then p is a NECESSARY condition for q
i don't understand what SUFFICIENT and NECESSARY mean in this case, how do ...
4
votes
3answers
154 views
Negating “Zach blocks e-mails and texts from Jennifer”
I am reviewing some basic propositional logic. The question that I have come across that has given some confusion is Zach blocks e-mails and texts from Jennifer where I am asked to find the negation ...
1
vote
2answers
42 views
Logic - Will a second parameter value inherit negation if the first parameter is false?
Will a second parameter value inherit negation if the first parameter is false?
Like:
(~A & B) → X
Is B false? Would it ...
2
votes
3answers
120 views
Are these two statement equivalent?
$\forall x \exists y P(x,y)$
$\exists x \forall y P(x,y)$
where P(x,y) means x is smaller than y.
I believe that they mean the same thing.
1
vote
3answers
98 views
Are these statements equivalent (quantifiers)?
$\neg \forall x \exists y \neg P(x,y)$ is equal to $\exists x \exists y \neg P(x,y)$
I had to make sure, because I wasn't sure at all.
0
votes
2answers
136 views
Discrete Math Logic Homework
Consider the following statement: $$\forall \epsilon > 0,\space\exists\delta>0:(|x-a|\lt\delta\implies|f(x)-L|\lt\epsilon).$$ (a) Write the converse of the statement.(b) Write the ...
