Post History
#4: Post edited
by
Chgg Clou
·
2022-05-29T03:12:05Z (over 2 years ago)
In "if and only if" proofs (biconditional logical equivalences), why's 1 direction easier to prove?
- In "if and only if" proofs, why's 1 direction easier to prove than the other?
[This list](https://math.stackexchange.com/q/3069488) on Math StackExchange instantiates equivalences where one direction can be proved effortlessly, but the other direction is racking to prove. If two propositions are equivalent, why can't they be proved with the same level of difficulty?- Please don't troll with frivolous "proofs" like
- [0 = 0 $\iff$ the riemann hypothesis is true](https://www.reddit.com/r/math/comments/3nrwbc/what_iff_statements_are_easy_to_prove_one_way_but/cvqvbzg/),- [a^n + b^n = c^n has integer solutions $\iff$ $n=2, \pm 1$](https://www.reddit.com/r/math/comments/3nrwbc/what_iff_statements_are_easy_to_prove_one_way_but/cvqr92l/), or- ['Fermat's last theorem' $\iff$ 0+0=0](https://redd.it/e2wtf7).[This Reddit thread](https://redd.it/e2wtf7) instances two common cases like [Hall's Theorem](https://old.reddit.com/r/math/comments/e2wtf7/are_you_aware_of_any_biconditional_statements/f8ysvyc/), and [a sequence of real numbers converges $\iff$ it is Cauchy](https://math.stackexchange.com/a/2170819) ($\Leftarrow$ is effortless, but $\Rightarrow$ is grueling).
- [This list](https://math.stackexchange.com/q/3069488) on Math StackExchange instantiates (biconditional) logical equivalences where one direction can be proved swimmingly, but the other direction is racking to prove. **If two propositions are equivalent, why can't they be proved with the same level of difficulty?**
- Please don't troll with frivolous "proofs" like
- - [0 = 0 $\iff$ The Riemann Hypothesis is true.](https://www.reddit.com/r/math/comments/3nrwbc/what_iff_statements_are_easy_to_prove_one_way_but/cvqvbzg/)
- - [$a^n + b^n = c^n$ has integer solutions $\iff$ $n=2, \pm 1$](https://www.reddit.com/r/math/comments/3nrwbc/what_iff_statements_are_easy_to_prove_one_way_but/cvqr92l/).
- - ['Fermat's last theorem' $\iff 0+0=0$](https://redd.it/e2wtf7).
- [This Reddit thread](https://redd.it/e2wtf7) instances two common cases like [Hall's Theorem](https://old.reddit.com/r/math/comments/e2wtf7/are_you_aware_of_any_biconditional_statements/f8ysvyc/), and [a sequence of complex numbers converges $\iff$ it is Cauchy](https://proofwiki.org/wiki/Cauchy%27s_Convergence_Criterion/Complex_Numbers) ($\Leftarrow$ is effortless, but $\Rightarrow$ is grueling).
#3: Post edited
by
Chgg Clou
·
2022-05-29T03:04:44Z (over 2 years ago)
In If and Only If Proofs, why's the proof for one direction easier than the other?
- In "if and only if" proofs (biconditional logical equivalences), why's 1 direction easier to prove?
- [This list](https://math.stackexchange.com/q/3069488) on Math StackExchange instantiates equivalences where one direction can be proved effortlessly, but the other direction is racking to prove. If two propositions are equivalent, why can't they be proved with the same level of difficulty?
Please don't troll with frivolous "proofs" like [0 = 0 iff the riemann hypothesis is true](https://www.reddit.com/r/math/comments/3nrwbc/what_iff_statements_are_easy_to_prove_one_way_but/cvqvbzg/), [a^n + b^n = c^n has integer solutions iff $n=2, \pm 1$](https://www.reddit.com/r/math/comments/3nrwbc/what_iff_statements_are_easy_to_prove_one_way_but/cvqr92l/), or ['Fermat's last theorem' iff 0+0=0](https://redd.it/e2wtf7). [This Reddit thread](https://redd.it/e2wtf7) instances two common cases like [Hall's Theorem](https://old.reddit.com/r/math/comments/e2wtf7/are_you_aware_of_any_biconditional_statements/f8ysvyc/), and [a sequence of real numbers converges iff it is Cauchy](https://math.stackexchange.com/a/2170819) ($\Leftarrow$ is effortless, but $\Rightarrow$ is grueling).
- [This list](https://math.stackexchange.com/q/3069488) on Math StackExchange instantiates equivalences where one direction can be proved effortlessly, but the other direction is racking to prove. If two propositions are equivalent, why can't they be proved with the same level of difficulty?
- Please don't troll with frivolous "proofs" like
- - [0 = 0 $\iff$ the riemann hypothesis is true](https://www.reddit.com/r/math/comments/3nrwbc/what_iff_statements_are_easy_to_prove_one_way_but/cvqvbzg/),
- - [a^n + b^n = c^n has integer solutions $\iff$ $n=2, \pm 1$](https://www.reddit.com/r/math/comments/3nrwbc/what_iff_statements_are_easy_to_prove_one_way_but/cvqr92l/), or
- - ['Fermat's last theorem' $\iff$ 0+0=0](https://redd.it/e2wtf7).
- [This Reddit thread](https://redd.it/e2wtf7) instances two common cases like [Hall's Theorem](https://old.reddit.com/r/math/comments/e2wtf7/are_you_aware_of_any_biconditional_statements/f8ysvyc/), and [a sequence of real numbers converges $\iff$ it is Cauchy](https://math.stackexchange.com/a/2170819) ($\Leftarrow$ is effortless, but $\Rightarrow$ is grueling).
#2: Post edited
by
Chgg Clou
·
2022-05-29T03:00:08Z (over 2 years ago)
[This list](https://math.stackexchange.com/q/3069488). [this question](https://math.stackexchange.com/q/3069488) substantiates proofs where one direction can be proved effortlessly, but the other direction is grueling to prove. Why? If two propositions are equivalent, wouldn't their proofs have the same level of difficulty?Please don't troll with frivolous "proofs" like ['Fermat's last theorem' iff 0+0=0](https://redd.it/e2wtf7), but that Reddit thread instances two common cases like [Hall's Theorem](https://old.reddit.com/r/math/comments/e2wtf7/are_you_aware_of_any_biconditional_statements/f8ysvyc/) and [a sequence of real numbers converges iff it is Cauchy](https://math.stackexchange.com/a/2170819). In latter, $\Leftarrow$ is effortless but $\Rightarrow$ is grueling.
- [This list](https://math.stackexchange.com/q/3069488) on Math StackExchange instantiates equivalences where one direction can be proved effortlessly, but the other direction is racking to prove. If two propositions are equivalent, why can't they be proved with the same level of difficulty?
- Please don't troll with frivolous "proofs" like [0 = 0 iff the riemann hypothesis is true](https://www.reddit.com/r/math/comments/3nrwbc/what_iff_statements_are_easy_to_prove_one_way_but/cvqvbzg/), [a^n + b^n = c^n has integer solutions iff $n=2, \pm 1$](https://www.reddit.com/r/math/comments/3nrwbc/what_iff_statements_are_easy_to_prove_one_way_but/cvqr92l/), or ['Fermat's last theorem' iff 0+0=0](https://redd.it/e2wtf7). [This Reddit thread](https://redd.it/e2wtf7) instances two common cases like [Hall's Theorem](https://old.reddit.com/r/math/comments/e2wtf7/are_you_aware_of_any_biconditional_statements/f8ysvyc/), and [a sequence of real numbers converges iff it is Cauchy](https://math.stackexchange.com/a/2170819) ($\Leftarrow$ is effortless, but $\Rightarrow$ is grueling).
#1: Initial revision
by
Chgg Clou
·
2021-03-09T05:27:22Z (over 3 years ago)
In If and Only If Proofs, why's the proof for one direction easier than the other?
[This list](https://math.stackexchange.com/q/3069488). [this question](https://math.stackexchange.com/q/3069488) substantiates proofs where one direction can be proved effortlessly, but the other direction is grueling to prove. Why? If two propositions are equivalent, wouldn't their proofs have the same level of difficulty? Please don't troll with frivolous "proofs" like ['Fermat's last theorem' iff 0+0=0](https://redd.it/e2wtf7), but that Reddit thread instances two common cases like [Hall's Theorem](https://old.reddit.com/r/math/comments/e2wtf7/are_you_aware_of_any_biconditional_statements/f8ysvyc/) and [a sequence of real numbers converges iff it is Cauchy](https://math.stackexchange.com/a/2170819). In latter, $\Leftarrow$ is effortless but $\Rightarrow$ is grueling.