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I am currently studying Discrete Maths in my 1st year.

Consider the following equivalences : (I'll just use absorption law as example)

In logical expression: p $\lor$ (p $\land$ q) $\equiv$ p

In set theory: P $\cup$ (P $\cap$ Q) = P

Every article I see uses only the "=" sign for laws of set identities.

I have tried searching for this, but to no avail. I also emailed my professor, in which he replied "the reasoning is simple, as the equivalence sign can only be used for logic expression." I don't quite understand this and would like to have a more intuitive approach.

My thoughts on this: the only possible reason that I could think of is because in set theory, the LHS and RHS are not completely "identical" to each other, as there are different ways of representing the LHS and RHS (for example, I could shade the venn diagrams differently). Whereas, in logical expressions, we are only interested in binary values T/F, as such, they are completely identical because there can only be "one" way of expressing the LHS and RHS.

I am not quite sure whether my trail of thoughts is clear... but I hope someone can share some light on this.

Asaf Karagila
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imaginarybuddy
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  • Equivalent sets are represented as '=' i think. – Aaa Lol_dude Oct 14 '21 at 02:35
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    Why are you picking on set theory? Why aren't you asking why we don't write $2+2\equiv4$ instead of $2+2=4$? – bof Oct 14 '21 at 02:43
  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Oct 14 '21 at 02:49
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    I think you're misunderstanding the usage of the symbol $\equiv$ in propositional logic. In mathematics, when we write $x+y=z$ or $x+y\lt z$ the $=$ or $\lt$ symbol is expressing a relation between the quantities $x+y$ and $z$. However, the symbol $+$ in $x+y$ is not expressing a relationship between $x$ and $y$, it is combining $x$ and $y$ to form a new quantity. Now, the $\equiv$ in $p\equiv q$ is not like $=$ or $\lt$, it is like $+$; that it, it's not expressing a relationship between the statements $p$ and $q$ but combining them to make a new statement. – bof Oct 14 '21 at 02:49
  • @bof: well to be pedantic it’s not clear whether the equivalence symbol here is meant syntactically (as material biconditional in the object language, which is like your $+$ analogy) or semantically (as semantic equivalence, a statement in the meta-language). Usually $\leftrightarrow$ is used for the former; in that case the $+$ analogy isn’t exactly right here. We aren’t combining well-formed formulas to get a new one; we’re stating something about them. – symplectomorphic Oct 14 '21 at 05:13
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    Equality between objects: numbers, sets, equivalence (same "meaning") between statements. – Mauro ALLEGRANZA Oct 14 '21 at 07:19
  • An identity $\equiv$ is a strong equality $=$, and a logical equivalence $\equiv$ is a strong equivalence $\Leftrightarrow$. – ryang Oct 17 '21 at 12:23

2 Answers2

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The idea is that when dealing with logical equivalence, two "statements" cannot be equal unless they are exactly the same statement. As mentioned by your instructor, you can indeed have logical equivalences between statements, but it does not make sense for them to be "equal"

However, with sets, you do in fact have equality. Recall that sets have elements, and two sets are equal if and only if they contain the exact same set of elements. If two sets are indeed equal, you will be shading the exact same portions of the Venn Diagram. If this does not happen, the sets are NOT equal.

One way that we define two sets to be equal in mathematics is as follows:

Let A and B be sets. We say $A=B$ if:

$$A\subseteq B$$ $$B\subseteq A$$

JJ Hoo
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By definition, two statements are equivalent exactly when they have the same truth value in all interpretations of their symbols.

Statements $p \lor (p \land q)$ and $p$ have the same truth value in all interpretations of literals $p,q$. So these statements are (logically) equivalent.

Although their evaluations are equal (in any particular interpretation), the statements are composed of different strings of symbols.

By definition, two sets are equivalent exactly when they have the same cardinality; that their is a one-to-one mapping between their elements.

This is much weaker than saying that their elements are identical; which is exactly when they are equal.

Sets $P\cup(P\cap Q)$ and $P$ will have exactly the same elements. So these sets are equal.

In short: Set theory and Logic theory just use different definitions for the terms "equality" and "equivalence".

Graham Kemp
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