Topology sets and functions

Related BrainMass Solutions

• Topologies

  is the topology having all the sets , which is an open set in . So . From the above discussion, we know the followings. 1. and 2. and 3. and do not contain each other. 4. and do not contain each other.

• Let X and Y be non-empty sets. If A_1 and A_2 are subsets of X, and B_1 and B_2 subsets of Y. Show that (A_1×B_1)U(A_2×B_2) = (A_1UA_2)×(B_1UB_2).

  Topology Sets and Functions (XLIV) Functions Let X and Y be non-empty sets.

• To show that the function f is an equivalence relation

  Topology Sets and Functions (XLV) Functions Let f : X --> Y be an arbitrary mapping.

• Let X and Y be non-empty sets and f a mapping of X into Y. Show that f is onto iff there exists a mapping h of Y into X such that fh = iX.

  Topology Sets and Functions (XXXVIII) Functions Let X and Y be non-empty sets and f a mapping of X into Y.

• Let X and Y be non-empty sets. If A_1 and A_2 are subsets of X, and B_1 and B_2 subsets of Y. Show that (A_1×B_1) intersection (A_2×B_2) = (A_1 intersection A_2)×(B_1 intersection B_2).

  Topology Sets and Functions (XLI) Functions Let X and Y be non-empty sets.

• Difference of the product of sets

  Topology Sets and Functions (XLII) Functions Let X and Y be non-empty sets.

• Let X and Y be non-empty sets and f a mapping of X into Y. Show that f is one-to-one iff there exists a mapping g of Y into X such that gf = iX.

  Topology Sets and Functions (XXXVII) Functions Let X and Y be non-empty sets and f a mapping of X into Y.

• Class of all finite unions of sets of the form A×B

  Topology Sets and Functions (XLIII) Functions Let X and Y be non-empty sets and let A and B be rings of subsets of X and Y respectively.

• The Symmetric Difference of two Sets: A ring of sets is a non-empty class A of sets such that if A and B are in A, then A difference B and A intersection B are also in A. Show that A must also contain the empty set.

  Topology Sets and Functions (XIV) The Algebra of Sets Ring of Sets The Symmetric Difference of two Sets