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Let $T$ and $S$ be two sets.

If for all elements $(x, y) \in T$ we also have $(x, y) \in S$, then it is not necessarily true that for all elements $(x, y) \in S$ we have $(x, y) \in T$.

Would this be correct?

Thank you.

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    what you are describing sounds like $T \subseteq S$. So No, this does not imply $S \subseteq T$. – 2017-01-20
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    Of course... an example of where it is true is when $T=S=\{(a,b)\}$ and an example of where it is not true is when $T=\{(a,b)\}$ and $S=\{(a,b),(c,d)\}$ – 2017-01-20

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Example: $T=\{(0,0)\}$ and $S=\{(0,0),(123,456)\}$.

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    I am a bit confused as to why this is correct. Does not the Op ask if it would be true for all elements, not just some? Why then is a specific example involving only a few elements considered an answer? Not trying to criticize just curious. – 2017-01-20
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    @user400188: Because this example is a witness that the claim in the question is true, which is equivalent to the existence of two sets $S,T$ for which it is not true that for all elements $(x,y)∈S$ we have $(x,y)∈T$. – 2017-01-20
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    Ah, I believe I read the question wrong. I saw "it is not necessarily true" as "is it not necessarily true" mistaking the it as an is. Funny how one word can negate the meaning in the English language. – 2017-01-21