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I'm stuck in a problem, need Help.

Associative property States that 

(a + b ) + c = a + (b + c)

Which is true

but what if i change the position of the numbers

(a + c) + b = a + (c + b)

Does it hold true for the associative Law because the answer will still be same..

Stuck in this problem. Need your help with proofs.

Thanks in Advance.

PrincessEev
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Harry47
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  • It's true if it's commutative. You're using the commutative property to commute the c and b. – N. Bar Jun 09 '20 at 03:42
  • Variables are just labels. In addition $(THING_1 + THING_2) + THING_3 = THING_1 + (THING_2 + THING_3)$ no matter what letter or labels you give them, and no matter what values the have. (Also you have commutative law too... if you really want to scramble them). – fleablood Jun 09 '20 at 03:44
  • N. Bar. It's true if it ISN'T commutative. It's just relabelling in the same position. Let $(a+b) + c=a+(b+c)$. So let $a'=a; c'=b; b'=c$ and you have $(a'+c') + b' = a' + (c' + b')$. It's just labels. – fleablood Jun 09 '20 at 03:46

1 Answers1

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Yep, it's true, even when you permute the symbols. Associativity means that $$(a + b) + c = a + (b + c)$$ for any $a, b, c$ in the appropriate set (real numbers perhaps?). It doesn't matter what I substitute in for these variables, this will always be true. Here are some true statements that follow from this: \begin{align*} (q + l) + v &= q + (l + v) \\ (1 + 3) + 5 &= 1 + (3 + 5) \\ (a + b) + (x + 1) &= a + (b + (x + 1)) \\ a^x + (b^y + c^z) &= (a^x + b^y) + c^z \\ a + (c + b) &= (a + c) + b \\ a + (a + a) &= (a + a) + a. \end{align*} Basically, you can substitute any expression you like in for $a, b, c$, so long as they evaluate to a real number, and you will get a true statement.

user797616
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  • (a + b) + c = a + (b + c ) agreed for this. But what if forthe same numbers i tell my students to go for this grouping as well i.e. (a + b) + c = a + (b + c ) which will be also equal to (a + c) + b – Harry47 Jun 09 '20 at 03:49
  • I'm not really sure what you're asking here, but in order to conclude that $(a + b) + c = (a + c) + b$, you'll need commutativity $b + c = c + b$ as well. – user797616 Jun 09 '20 at 03:52
  • but i want clarity on this point that in associative law for three numbers a,b,c we have (a+b)+c= a+(b+c ) + b but if student writes another option along with these two which is (a+c) +b would it be justified under associative law. – Harry47 Jun 09 '20 at 04:00
  • Well, $(a + b) + c \neq a + (b + c) + b$, unless $b = 0$. No, it is not justified, purely by associative law, to say $(a + c) + b$ is equal to either $(a + b) + c$ or $a + (b + c)$. It is justified with associative and commutative laws together, but not just associative law. – user797616 Jun 09 '20 at 04:11