Hello, I am having trouble interpreting the definition of the union of two sets as given in Modern Abstract Algebra in Schaum's Outlines. I can see by example but I can't seem to interpret the definition. Could someone reword this for me or give me another spin on this definition? Thankyou. Defintion as in Modern Abstract Algebra in Schaum's Outlines Let A and B be given sets. The set of all elements which belong to A alone or to B alone or to both A and B is called the union of A and B. I understand the example: Let A = {1,2,3,4} and B = {2,3,5,8,10}; then A union B = {1,2,3,4,5,8,10} And the way I interpret the union of two sets is this: Given two sets A and B, let the union of A and B be C. Then C contains the following: Elements common to both A and B. Elements in A and not in B. And elements in B but not in A. But I don't get the definition as given by Schaums. Any help is appreciated. Thanks again.
I don't really understand what the difference is between your definition and Schaums. You're both saying the same thing.
I don't know. Perhaps I am misinterpreting Schuam's definition. I don't see the equivalence between Schaum's definition and mine. I interpret Schuams definition as follows: The set of all elements which belong to A alone. == The set consisting of just the elements of A. Meaning if the set consists of just the elements of A then B is an empty set. A union B = A The set of all elements which belong to B alone == The set consisting of just the elements of B. Similarly if the set consists of just the elements of B then A must be an empty set. B union A = B The set of all elements which belong to both A and B == The set consisting of the elements in both A and in B. This last part I can see.
u(AUB) = u(A) + u(B) - u(ANB) [The 'N' is intersect of A and B, in real life it looks like an upside-down U.] The two different cases are whether A and B are discrete sets or not. If A and B do not have anything in commmon, there is no intersection, hence u(ANB) = {} = 0, the empty set.
Could you elaborate on your notation? I don't think I have seen the notation u(AUB) before. Just what does the u(...) stand for? I understand that if A and B have nothing in common that their intersection is the empty set. I don't see the connection though.
Perhaps this can be of some assistance: [tex]A \cup B = \lbrace e \mid e \in A \lor e \in B \rbrace[/tex] The size of the union would be [tex]|A \cup B| = |A| + |B| - |A \cap B|[/tex] Whch means "all the elements in [tex]A[/tex] + all the elements in [tex]B[/tex] - the elements in both [tex]A[/tex] and [tex]B[/tex]". Nille
You can easily see from attachment that Union means The no. of ekements in both sets without repeating the same no in both sets u can easily see Union will be RED+BLUE-GREY this is because as i say not to repeat the common elements we subtract the Grey portion once coz when we add RED+BLUE they common elements are added up twice so we have to delete one
sumset a more general union is this. let T be a collection of sets, usually at least two sets. for example, T={A,B}. [tex]\bigcup T[/tex] is the set of elements in at least one member of T. [tex]\bigcup \left\{ A,B\right\} =A\cup B[/tex] is the set of elements in at least one of A and B. ie, if x is in A or B (or both), then x is in the union. if T had three sets in it, the same definition would apply: x is in the union of three sets if it is in at least one of the sets in T.
Thanks again everyone. These many different perspectives has given me a better understanding of the def. of union of sets.