# Equivalence relations, operator

$\def\op{\mathbin{\#}}\def\R{\mathbin R}$Original question:

A logical operation between two propositions $p$ and $q$ is denoted as $p\op q$. It is only true when $p$ is true and $q$ is false;otherwise it is false. Let $R$ be a relation on the set of ordered pairs of propositions, where $R$ is defined as follows:

$\bigl((a,b),(c,d)\bigr) ∈ R$, if and only if $(a \land b) \op (c \vee d)$ is true.

Is R an equivalence relation?

For symmetric, the rule is $∀a,b ∈ B$ s.t. $a\R b$, $b\R a$ is true.

But when I do the relation, do I bring the operator (i.e. $\land$ and $\lor$) over also?

For example:

1) $(a \land b) \op (c \lor d) \to (c \lor d) \op (a \land b)$

2) $(a \land b) \op (c \lor d) → (c \land d) \op (a \lor b)$

which is the correct way of representing?

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The part that says "Original question" doesn't actually contain a question. Also, the title mentions equivalence relations, but the body doesn't. It seems that perhaps there's something missing at the end of the block quote where the question asks something like whether $R$ is an equivalence relation? – joriki Nov 26 '12 at 13:16

(a) A relation $R$ on the set $B$, is symmetric if and only if $\forall a,b \in B$, if $\;a\, R\, b$ is true, then $b\,R\,a$ is true.

Here the elements of $B$ are ordered pairs of propositions.

$\bigl((a,b),(c,d)\bigr) \in R$, if and only if $(a \land b) \op (c \vee d)$ is true.

So you must show that $R$ is symmetric, you need to show that $(2)$, as you've written, holds. That is, you need to show that $$\forall (a, b) \in B,\; [(a \land b) \op (c \lor d)] \rightarrow [(c \land d) \op (a \lor b)].$$

Whether your relation is symmetric or not depends on the set on which the relation is defined. If it is defined on the set of all ordered pairs of all propositions, e.g., $(a, b)$, where $a, b\, \in \{\text{All possible propositions}\}$, then you should be able to easily find a pair of propositions that provides a counterexample to symmetry.

Note: that to determine whether $R$ is an equivalence relation on the set $B$ of pairs $(x, y)$, you need to also need to determine whether the relation is reflexive and transitive, as well as symmetric:

(b) The relation $R$ on the set $B$ is reflexive if and only if for all $(a,b) \in B$, $(a \land b) \op (a \lor b)$ is true.

(c) The relation $R$ is transitive if and only if for all $(a, b), (b, c), (c, d) \in B$, if $(a \land b) \op (b \lor c)$ and $(b \land c) \op (c \lor d)$ are both true, then $(a \land b) \op (c \lor d)$ is also true.

Note that for $\op$, as defined, $(p \op q) \iff (p \land \lnot q) \iff \lnot(p\rightarrow q)$.

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+1 YOU R LOVIN LOGIC. ;-) – Babak S. Aug 27 '13 at 5:58

Way 2 is the correct way, of course.
Arguments of R are pairs of propositions, not the operations $\wedge$ and $\vee$ on that pairs.
So, R is not symmetric.

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