What is the relation A = B = C called in a proof?

Question

When writing a proof if I have the relationship $$ A = B = C $$

And I want to use that to prove $$ A = C $$

I remember there being some term for it. What is that term, and what would be an appropriate intertext in this situation?

The obvious answer is transitivity, but I think a bit of caution is due. In normal practice one uses transitivity implicitly (i.e., no intertext). In formal proofs, it depends on the axioms and prior theorems given on equality. In fact, the question looks like the OP is actually trying to prove that $=$ is transitive, so assuming it's transitive could be circular... — askyle, Jun 12 '14 at 09:45

mirichan · Accepted Answer · 2014-06-12T02:51:52.450

9

What you're looking for is the transitive property of equality. http://www.mathwords.com/t/transitive_property.htm

$$ \text{Given}\\ A = B\\ B = C\\ \text{Equality is transitive}\\ \therefore A = C $$

edited Jun 12 '14 at 02:51

answered Jun 12 '14 at 02:45

mirichan

score 4 · Answer 2 · answered Jun 12 '14 at 02:44

4

Transitivity is the word you're looking for.

answered Jun 12 '14 at 02:44

beep-boop

score 3 · Answer 3 · answered Jun 12 '14 at 02:44

3

" $A=B, \ B=C \implies A=C$ " is called transitivity of equality.

answered Jun 12 '14 at 02:44

lhf

3 Answers3