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2answers
49 views

How to find the Equivalence class for a given set?

I'm really having trouble understanding these equivalence classes. Could someone please guide me through the following problem step by step, and help explain this. I have a final exam next week, and ...
0
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2answers
37 views

Inference Proof with Quantifiers

I am trying to figure out this implication proof. Can any of you guys tell me how to prove this? Prove ∀x((¬P(x) ∧ Q(x)) → R(x)) Implies ∀x(¬R(x) → P(x))
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1answer
65 views

Propositional Logic with rules of inference problem.

$$ \begin{array}{l} 1.\>\>\>\> (r ∧ ¬s) ∨ (q ∧ ¬s)\\ 2.\>\>\>\> ¬s → ((p ∧ r) → u)\\ 3.\>\>\>\> u → (s ∧ ¬t)\\ ...
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votes
1answer
45 views

What is the proper way to format a hypothetical syllogism proof?

Problem: Show that these three statements are equivalent, where $a, b \in R:$ (i) $a < b$, (ii) the average of $a, b,$ is greater than $a,$ and (iii) the average of $a$ and $b$ is less than $b$. ...
0
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3answers
98 views

Can a premise imply contradictory statements?

Can a premise imply contradictory statements? Can two contradictory premises imply the same conclusion? Determine the answers to these questions by doing the following. Prove or disprove: the ...
2
votes
1answer
29 views

Generated equivalence relations in logics

Let $L$ be some logic (FO or stronger which is not important for this purpose). Given a $\tau$-structure $A$ and a formula $\varphi(x_1, \dots x_n) \in L[\tau]$ with free variables $x_1, \dots, x_n$. ...
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votes
1answer
39 views

Help needed with an equivalence relation task on natural numbers

I'm having a bit difficulties understanding and solving this task. I would appreciate any help on how you can solve tasks like this. Here is the task: Let ~ be an equivalence relation on the ...
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0answers
27 views

How to prove an equivalence relation [duplicate]

I have a set of tasks which is suggested done for understanding this weeks lesson in logic, and more specifically partitional logic. I have one more task left before I am done with the set, but I ...
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1answer
154 views

Proving the symmetry of an equivalence relation

When proving the symmetry of an equivalence relation, must each equivalence class be closed under symmetry. for example: the relation both x and y > 10 or both x and y < 10 across all ...
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1answer
276 views

Proving that a pair of equivalence classes must be identical or disjoint

Give an equivalence $R$ relation over a set $A$: $$C_x=\{y\in{A}:xRy\}$$ I'm trying to prove that if $x,y\in{A}$, Either $C_x=C_y$ or $C_x\cap{C_y}=\{\}$. In other words, $C_x$ and $C_y$ must be ...
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1answer
72 views

Logic: Equivalence Relation

Let $\Gamma$ be a maximally consistent set of formulas of $\mathcal{L}$. For any two terms $\tau_1$ and $\tau_2$ of $\mathcal{L}$, define that $\tau_1 \cong_\Gamma \tau_2$ if and only if ...
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1answer
131 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\; ...
3
votes
3answers
107 views

Confusion on understanding a proposition on equivalence classes

I am given to prove this proposition on equivalence classes. Each element of $A$ is an element of one and only one equivalent class. The part that is confusing is one and only one. It sounds ...
3
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2answers
238 views

Would this relation be an equivalence relation?

I am a bit stuck on this one question from my homework and for some reason it isn't making any sense to me. I would really appreciate it if somebody could explain it to me how I can go about to ...
1
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1answer
127 views

Question relating equivalence relations/classes

So, for a course I'm following we got some practice exams to prepare for the finals. However, for some of these we do not have answers, nor can I find someone who is certain of his answer on the ...