Prove if a relation is left-total and right-uniqueIn this exercise, we are dealing with Tuple sets.So let M ⊂ P ( A × N ) be the set of all legal tuple sets. P (X) denotes the power set of a set X. Now, Let the tuple set operation be defined as + so that +: M × M → M.Prove or disprove that + is left-total and right-unique.These types of relations are uncommon in English, it seems. So, it's a bit difficult for me to understand how to prove it.To my knowledge, a left-total relation means that, e.g. if ∀ x ∈ A ∃ y ∈ N ( x , y ) ∈ M, i.e. for every x ∈ A there exists (at least) one y ∈ N, so that ( x , y ) ∈ M.

Question

Prove if a relation is left-total and right-uniqueIn this exercise, we are dealing with Tuple sets.So let   M  ⊂  P  (  A  ×      N    ) be the set of all legal tuple sets. P (X) denotes the power set of a set X. Now, Let the tuple set operation be defined as   + so that   +:   M  ×  M  →  M.Prove or disprove that + is left-total and right-unique.These types of relations are uncommon in English, it seems. So, it&#039;s a bit difficult for me to understand how to prove it.To my knowledge, a left-total relation means that, e.g. if   ∀  x  ∈  A   ∃  y  ∈  N   (  x  ,  y  )  ∈  M, i.e. for every   x  ∈  A there exists (at least) one   y  ∈  N, so that   (  x  ,  y  )  ∈  M.

Maggie Bowman · Accepted Answer

Step 1Note that   + is a function, and functions are just special cases of relations.Hence, for left-totality of   +, you wish to show that for any pair   (      m    1    ,      m    2    )  ∈  M  ×  M, that we can find   m  ∈  M where       m    1    +      m    2    =  m (to use the usual conventions for operations). That is, essentially, left-totality for function relations amounts to ensuring each input has an output.Step 2At present, this is not satisfied. How would you define       m    1    +      m    2   for two tuples of different sizes? (I assume we add them in the &quot;usual&quot; way, namely pointwise.) For instance, if   (  1  ,  2  )  ,  (  1  ,  2  ,  3  ,  4  )  ∈  M, how would   (  1  ,  2  )  +  (  1  ,  2  ,  3  ,  4  ) be defined? In fact even if M is limited to tuples of the same size, you won&#039;t necessarily have the desired property unless A is closed under addition.Right-uniqueness is another property expected of functions (notice how these two relate? We&#039;re showing + is a function!); namely, any input goes to at most one output. Whenever the addition of tuples is defined, then it inherits this property from that of the addition of elements of + (if present).