Theorems in the form of “if and only if” such that the proof of one direction is extremely EASY to prove...












4














I believe this is a common phenomenon in mathematics, but surprisingly, no such list has been created on this site. I don't know if it's of value, just out of curiosity, I want to see more examples. So my request is:



Theorems in the form of "if and only if" that the proof of one direction is extremely EASY, or intuitive, or make use of some standard techniques (e.g. diagram chasing), while the other one is extremely HARD, or counterintuitive, or require a certain amount of creativity. The contrast should be as apparent as possible.



Thanks in advance.










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  • One of my college professors likened this phenomenon to running one's palm along one's face. Down is easy; up is not.
    – Blue
    2 hours ago








  • 1




    That's a good metaphor!
    – YuiTo Cheng
    2 hours ago






  • 2




    The reason nobody has created such a list may be the fact that most (nontrivial) "iff" theorems belong on your list. It would be more interesting to see a list of "iff" theorems where both directions are nontrivial.
    – bof
    2 hours ago










  • @bof I've mentioned I don't know if this question is of any value. I just think this phenomenon is intriguing, and want to know more of that nature.
    – YuiTo Cheng
    1 hour ago


















4














I believe this is a common phenomenon in mathematics, but surprisingly, no such list has been created on this site. I don't know if it's of value, just out of curiosity, I want to see more examples. So my request is:



Theorems in the form of "if and only if" that the proof of one direction is extremely EASY, or intuitive, or make use of some standard techniques (e.g. diagram chasing), while the other one is extremely HARD, or counterintuitive, or require a certain amount of creativity. The contrast should be as apparent as possible.



Thanks in advance.










share|cite|improve this question






















  • One of my college professors likened this phenomenon to running one's palm along one's face. Down is easy; up is not.
    – Blue
    2 hours ago








  • 1




    That's a good metaphor!
    – YuiTo Cheng
    2 hours ago






  • 2




    The reason nobody has created such a list may be the fact that most (nontrivial) "iff" theorems belong on your list. It would be more interesting to see a list of "iff" theorems where both directions are nontrivial.
    – bof
    2 hours ago










  • @bof I've mentioned I don't know if this question is of any value. I just think this phenomenon is intriguing, and want to know more of that nature.
    – YuiTo Cheng
    1 hour ago
















4












4








4







I believe this is a common phenomenon in mathematics, but surprisingly, no such list has been created on this site. I don't know if it's of value, just out of curiosity, I want to see more examples. So my request is:



Theorems in the form of "if and only if" that the proof of one direction is extremely EASY, or intuitive, or make use of some standard techniques (e.g. diagram chasing), while the other one is extremely HARD, or counterintuitive, or require a certain amount of creativity. The contrast should be as apparent as possible.



Thanks in advance.










share|cite|improve this question













I believe this is a common phenomenon in mathematics, but surprisingly, no such list has been created on this site. I don't know if it's of value, just out of curiosity, I want to see more examples. So my request is:



Theorems in the form of "if and only if" that the proof of one direction is extremely EASY, or intuitive, or make use of some standard techniques (e.g. diagram chasing), while the other one is extremely HARD, or counterintuitive, or require a certain amount of creativity. The contrast should be as apparent as possible.



Thanks in advance.







soft-question big-list






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asked 2 hours ago









YuiTo ChengYuiTo Cheng

154112




154112












  • One of my college professors likened this phenomenon to running one's palm along one's face. Down is easy; up is not.
    – Blue
    2 hours ago








  • 1




    That's a good metaphor!
    – YuiTo Cheng
    2 hours ago






  • 2




    The reason nobody has created such a list may be the fact that most (nontrivial) "iff" theorems belong on your list. It would be more interesting to see a list of "iff" theorems where both directions are nontrivial.
    – bof
    2 hours ago










  • @bof I've mentioned I don't know if this question is of any value. I just think this phenomenon is intriguing, and want to know more of that nature.
    – YuiTo Cheng
    1 hour ago




















  • One of my college professors likened this phenomenon to running one's palm along one's face. Down is easy; up is not.
    – Blue
    2 hours ago








  • 1




    That's a good metaphor!
    – YuiTo Cheng
    2 hours ago






  • 2




    The reason nobody has created such a list may be the fact that most (nontrivial) "iff" theorems belong on your list. It would be more interesting to see a list of "iff" theorems where both directions are nontrivial.
    – bof
    2 hours ago










  • @bof I've mentioned I don't know if this question is of any value. I just think this phenomenon is intriguing, and want to know more of that nature.
    – YuiTo Cheng
    1 hour ago


















One of my college professors likened this phenomenon to running one's palm along one's face. Down is easy; up is not.
– Blue
2 hours ago






One of my college professors likened this phenomenon to running one's palm along one's face. Down is easy; up is not.
– Blue
2 hours ago






1




1




That's a good metaphor!
– YuiTo Cheng
2 hours ago




That's a good metaphor!
– YuiTo Cheng
2 hours ago




2




2




The reason nobody has created such a list may be the fact that most (nontrivial) "iff" theorems belong on your list. It would be more interesting to see a list of "iff" theorems where both directions are nontrivial.
– bof
2 hours ago




The reason nobody has created such a list may be the fact that most (nontrivial) "iff" theorems belong on your list. It would be more interesting to see a list of "iff" theorems where both directions are nontrivial.
– bof
2 hours ago












@bof I've mentioned I don't know if this question is of any value. I just think this phenomenon is intriguing, and want to know more of that nature.
– YuiTo Cheng
1 hour ago






@bof I've mentioned I don't know if this question is of any value. I just think this phenomenon is intriguing, and want to know more of that nature.
– YuiTo Cheng
1 hour ago












5 Answers
5






active

oldest

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3














"A compact 3-manifold is simply-connected if and only if it is homeomorphic to the 3-sphere."



One direction [only if] is the Poincaré conjecture. The other direction shouldn't be too bad hopefully.






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    1














    The bisectors of two angles of a triangle are of equal length if and only if the two bisected angles are equal. If the two angles are equal, the triangle is isosceles and the proof is very easy. However proving that a triangle must be isosceles if the bisectors of two of its angles are of equal length seems to be quite difficult.






    share|cite|improve this answer








    New contributor




    lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      0














      The product of topological spaces is compact if and only if all of them are compact. One direction is trivial because the projections are continuous and surjective functions. The another direction, well, is the axiom of choice.






      share|cite|improve this answer





























        0














        Let $n$ be a positive integer. The equation $x^n+y^n=z^n$ is solvable in positive integers $x,y,z$ iff $nle2$.






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          All planar graphs are $n$-colorable iff $nge4$.






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            5 Answers
            5






            active

            oldest

            votes








            5 Answers
            5






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            3














            "A compact 3-manifold is simply-connected if and only if it is homeomorphic to the 3-sphere."



            One direction [only if] is the Poincaré conjecture. The other direction shouldn't be too bad hopefully.






            share|cite|improve this answer


























              3














              "A compact 3-manifold is simply-connected if and only if it is homeomorphic to the 3-sphere."



              One direction [only if] is the Poincaré conjecture. The other direction shouldn't be too bad hopefully.






              share|cite|improve this answer
























                3












                3








                3






                "A compact 3-manifold is simply-connected if and only if it is homeomorphic to the 3-sphere."



                One direction [only if] is the Poincaré conjecture. The other direction shouldn't be too bad hopefully.






                share|cite|improve this answer












                "A compact 3-manifold is simply-connected if and only if it is homeomorphic to the 3-sphere."



                One direction [only if] is the Poincaré conjecture. The other direction shouldn't be too bad hopefully.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 2 hours ago









                Alvin JinAlvin Jin

                2,127918




                2,127918























                    1














                    The bisectors of two angles of a triangle are of equal length if and only if the two bisected angles are equal. If the two angles are equal, the triangle is isosceles and the proof is very easy. However proving that a triangle must be isosceles if the bisectors of two of its angles are of equal length seems to be quite difficult.






                    share|cite|improve this answer








                    New contributor




                    lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.























                      1














                      The bisectors of two angles of a triangle are of equal length if and only if the two bisected angles are equal. If the two angles are equal, the triangle is isosceles and the proof is very easy. However proving that a triangle must be isosceles if the bisectors of two of its angles are of equal length seems to be quite difficult.






                      share|cite|improve this answer








                      New contributor




                      lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                      Check out our Code of Conduct.





















                        1












                        1








                        1






                        The bisectors of two angles of a triangle are of equal length if and only if the two bisected angles are equal. If the two angles are equal, the triangle is isosceles and the proof is very easy. However proving that a triangle must be isosceles if the bisectors of two of its angles are of equal length seems to be quite difficult.






                        share|cite|improve this answer








                        New contributor




                        lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                        Check out our Code of Conduct.









                        The bisectors of two angles of a triangle are of equal length if and only if the two bisected angles are equal. If the two angles are equal, the triangle is isosceles and the proof is very easy. However proving that a triangle must be isosceles if the bisectors of two of its angles are of equal length seems to be quite difficult.







                        share|cite|improve this answer








                        New contributor




                        lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                        Check out our Code of Conduct.









                        share|cite|improve this answer



                        share|cite|improve this answer






                        New contributor




                        lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                        Check out our Code of Conduct.









                        answered 33 mins ago









                        lonza leggieralonza leggiera

                        1534




                        1534




                        New contributor




                        lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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                        New contributor





                        lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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                        lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                        Check out our Code of Conduct.























                            0














                            The product of topological spaces is compact if and only if all of them are compact. One direction is trivial because the projections are continuous and surjective functions. The another direction, well, is the axiom of choice.






                            share|cite|improve this answer


























                              0














                              The product of topological spaces is compact if and only if all of them are compact. One direction is trivial because the projections are continuous and surjective functions. The another direction, well, is the axiom of choice.






                              share|cite|improve this answer
























                                0












                                0








                                0






                                The product of topological spaces is compact if and only if all of them are compact. One direction is trivial because the projections are continuous and surjective functions. The another direction, well, is the axiom of choice.






                                share|cite|improve this answer












                                The product of topological spaces is compact if and only if all of them are compact. One direction is trivial because the projections are continuous and surjective functions. The another direction, well, is the axiom of choice.







                                share|cite|improve this answer












                                share|cite|improve this answer



                                share|cite|improve this answer










                                answered 57 mins ago









                                Carlos JiménezCarlos Jiménez

                                2,3551519




                                2,3551519























                                    0














                                    Let $n$ be a positive integer. The equation $x^n+y^n=z^n$ is solvable in positive integers $x,y,z$ iff $nle2$.






                                    share|cite


























                                      0














                                      Let $n$ be a positive integer. The equation $x^n+y^n=z^n$ is solvable in positive integers $x,y,z$ iff $nle2$.






                                      share|cite
























                                        0












                                        0








                                        0






                                        Let $n$ be a positive integer. The equation $x^n+y^n=z^n$ is solvable in positive integers $x,y,z$ iff $nle2$.






                                        share|cite












                                        Let $n$ be a positive integer. The equation $x^n+y^n=z^n$ is solvable in positive integers $x,y,z$ iff $nle2$.







                                        share|cite












                                        share|cite



                                        share|cite










                                        answered 5 mins ago









                                        bofbof

                                        50.5k457119




                                        50.5k457119























                                            0














                                            All planar graphs are $n$-colorable iff $nge4$.






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                                              0














                                              All planar graphs are $n$-colorable iff $nge4$.






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                                                0












                                                0








                                                0






                                                All planar graphs are $n$-colorable iff $nge4$.






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                                                All planar graphs are $n$-colorable iff $nge4$.







                                                share|cite












                                                share|cite



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                                                answered 3 mins ago









                                                bofbof

                                                50.5k457119




                                                50.5k457119






























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