# Names for laws involving implication and/or exponentiation

The following are all logical equivalences

1. $p \wedge q \Rightarrow r \;=\; p \Rightarrow (q \Rightarrow r)$
2. $p \Rightarrow q \wedge r \;=\; (p \Rightarrow q) \wedge (p \Rightarrow r)$
3. $p \vee q \Rightarrow r \;=\; (p \Rightarrow r) \wedge (q \Rightarrow r)$

Do they individually have names?

There are corresponding laws of exponentiation. As far as I know they don't have individual names.

1. $r^{pq} \;=\; (r^p)^q$
2. $(qr)^p \;=\; q^p \, r^p$
3. $r^{(p+q)} \;=\; r^p \, r^q$
• I think this might be better to ask on MSE since this isn't really about math education. – user6648 Jan 6 '17 at 14:22
• @tilper You are probably right. My motivation was that I'm teaching logic, so I was wondering what other teachers teach. So I thought of this site first. – Theodore Norvell Jan 6 '17 at 21:23
• The first three have nice interpretations in terms of category theory: they're all deduced from a special case of the tensor-Hom adjunction, also known in programming as "currying". However, this probably isn't something you'd mention in an introductory logic course. (Not posting as an answer because (1) it doesn't really answer the question and (2) this probably does belong on MSE.) – Daniel Hast Jan 7 '17 at 1:17
• I've seen the first one called "shunting" in the context of propositional logic and "currying" in the context of type theory. – Theodore Norvell Jan 8 '17 at 13:01

For exponentiation, we might say that

1. $(qr)^p \;=\; q^p \, r^p$
2. $r^{(p+q)} \;=\; r^p \, r^q$

are distributive laws.

And (perhaps) that

1. $r^{pq} \;=\; (r^p)^q$

is an associative law

• I wouldn't call $r^{pq} = (r^p)^q$ an associative law — associativity is when parentheses can be moved or omitted (because the value is the same regardless of how the expression is parenthesized), while with this here, one of the operations has to be changed (from multiplication to exponentiation) when the parentheses are moved. – Daniel Hast Jan 7 '17 at 23:09
• I like "distributive laws". Also "distributive law" works for the other two ways of distributing conjunction and disjunction over implication. I rather agree with Daniel about "associative law". – Theodore Norvell Jan 8 '17 at 19:31

The names for your exponentiation laws are pretty standardized in elementary algebra texts; for example, from Martin-Gay, Prealgebra & Introductory Algebra, Sec 10.1 (ordering as per the book):

1. Product Rule (for Exponents): $a^m \cdot a^n = a^{m+n}$
2. Power Rule (for Exponents): $(a^m)^n = a^{mn}$
3. Power of a Product Rule: $(ab)^n = a^nb^n$

Looking at other sources: Bittinger's Intermediate Algebra (Sec. R.7): (1) Product Rule, (2) Power Rule, (3) Raising a Product to a Power.

Ratti & McWaters, Precalculus: A Right Triangle Approach (Sec. P.2): (1) Product Rule of Exponents, (2) Power Rule for Exponents, (3) Power-of-a-Product Rule.

Now, it's common to call standard distribution of real numbers $a(b+c) = ab + ac$ by the fully-formed name of the "Distributive Property of Multiplication Over Addition" (e.g., Martin-Gay does this); so I think it's also fair to call (3) the "Distribution Property of Exponents Over Multiplication", and I've seen something like that in places.

In my college algebra classes, I consolidate these rules into what I call the "General Distribution Rule": operations distribute over any operation one line lower in the order-of-operations. But that's a phrasing of my own invention, as far as I know.

(Looking at a significantly older text, Rietz and Crathorne, Introductory College Algebra (1933), Ch. 1, they say of $a^ma^n = a^{m+n}$, "This is often called the first law of exponents". Names for the other rules are not clearly given. I doubt this is common usage today; but for what it's worth, the ordering in all of these books referenced is identical, with the Product Rule indeed always given first.)

• I've never heard the terms "power rule for exponents", "power of a product rule", "product rule for exponents", or "first law of exponents"; I suspect most mathematicians wouldn't recognize these, either, though at least "power of a product rule" is a clear enough name that I could guess what it means. "Distributive property of [one operation] over [another operation]" is the only one here that's in common usage as far as I know. – Daniel Hast Jan 7 '17 at 23:04
• @DanielHast: I agree that I went through all my undergraduate and graduate school without ever thinking to have names for these identities. But I've found that the names in elementary algebra texts are actually pretty standardized (I added two more references above). If one wanted to Google these topics, those names would pop up appropriate resources (e.g., "product rule of exponents" gives 364K hits and an auto-answer snippet in Google). – Daniel R. Collins Jan 7 '17 at 23:21
• Daniel Collins is completely correct that these names are pretty universal in the context of secondary education in the United States, and @DanielHast is equally correct that they are almost entirely unknown outside of that context (with the possible exception of remedial "college algebra" courses, especially at the Community College level, which in many ways resemble secondary education more than post-secondary). – mweiss Jan 8 '17 at 18:08
• @mweiss: I can't help but point out that "college algebra" is universally not a remedial course at community colleges; rather, it's referred to as a "gateway" course, i.e., first transferable credit course. Maybe you meant that metaphorically. – Daniel R. Collins Feb 19 '17 at 17:56
• @DanielR.Collins Point taken, although I am not sure "remedial" and "transferable" are mutually exclusive. To me "remedial" implies remediation, i.e. a re-teaching of material that was taught (but not learned in, or not remembered from) previous classes at the high school level. Do you disagree that college algebra courses at the CC level can be accurately described that way? – mweiss Feb 19 '17 at 23:23