Proving the Existence of b: B-->A for Tricky Algebra Proof

In summary, the conversation discusses a tricky algebra proof involving mappings and the need to prove the existence of a specific mapping. The key idea is that the proof can be approached by showing that R is injective.
  • #1
Pearce_09
74
0
tricky algebra proof...

hello,

consider the mappings: (R: A-->B)

Suppose T: C-->A and S: C-->A satisfy RT = RS then T = S
prove that there exists a; b: B-->A such that bR = idA (identity of A)

well, I am not sure if I can say that b (inverse) = R
since b maps B to A .. and R maps A to B... and if i can say that...
how do i approach the proof?

I know that RT = RS then T = S.. should i work with this.? using b (inverse)...
or should i try to see if R or b is injective?...

ie. b(inverse)T = b(inverse)S ... ??
 
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  • #2
The idea of left cancellable (RS=RT => S=T) is exactly the same as R being injective.
 

FAQ: Proving the Existence of b: B-->A for Tricky Algebra Proof

How do you prove the existence of b in a tricky algebra proof?

The existence of b can be proven by using mathematical induction. This involves showing that the statement holds true for a base case, and then using a proof by contradiction to show that it also holds true for any other case.

Can you provide an example of proving the existence of b in a tricky algebra proof?

Sure, let's say we are trying to prove that the equation 2n + 1 = 3 has a solution for n. We can start by showing that it holds true for n = 1, since 2(1) + 1 = 3. Then, we assume that it also holds true for some arbitrary k, and use a proof by contradiction to show that it also holds true for k + 1. This proves that the equation has a solution for all positive integers, thus proving the existence of b.

What is the significance of proving the existence of b in a tricky algebra proof?

Proving the existence of b is important because it allows us to show that a statement or equation holds true for all possible cases, not just a few specific ones. This strengthens the validity of our proof and ensures that it is applicable in all scenarios.

Are there any other methods for proving the existence of b besides mathematical induction?

Yes, there are other methods such as direct proof, proof by contradiction, and proof by contrapositive. However, mathematical induction is often the most efficient and effective method for proving the existence of b in tricky algebra proofs.

What are some common mistakes to avoid when trying to prove the existence of b in a tricky algebra proof?

One common mistake is assuming that the statement holds true for a specific case, but not considering all other cases. Another mistake is using circular reasoning, where the statement being proven is used as a premise in the proof. It is important to carefully consider all cases and use logical and valid reasoning in the proof.

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