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
    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


















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
    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
















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
    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




















  • 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
    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


















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
1 hour 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
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






@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












3 Answers
3






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








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      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.






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        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        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 26 mins ago









                    lonza leggieralonza leggiera

                    1534




                    1534




                    New contributor




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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.
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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














                          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 49 mins ago









                            Carlos JiménezCarlos Jiménez

                            2,3551519




                            2,3551519






























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