Constant Map

A mapping %:A->B is called a constant map if there exists b.(b not) belonging to B such that %(a) = b. for all a belonging to A. Show that a mapping %:A->B is constant if and only if %$=% for all $:A->A

Proof by Mathematical Induction : Planes, Lines and Regions

Suppose that n straight lines in the plane are positioned so that no two are parallel an no three pass throught the same point. Show that they divide the plane into 1/2(n^2 + n + 2) distinct regions.

Well-Ordering Axiom - Strong Induction

Prove the well-ordering Axiom by strong induction.

Proof by Induction

Let pn denote the statement: "In any class of n algebra students, every student obtains the same grade." Then p1 is clearly true. If pn is satisfied for n>1, suppose that x1, x2, ...., xn all have the same grade (by induction) as do x2, x3,....,xn+1. Thus x1,x2,...,xn+1 all have the same grade (the same as xn), so pn+1 is true. ...continues

Binary Operations : Cayley Table

Consider the Cayley table: (see file) Show that there is only one way to complete table (1) so that the resulting operation is associative, and that the result makes {a,b} into a commutative monoid.

Binary Operations : Idempotence

An element e of a monoid M is called an idempotent if e^2 = e. If M is finite, show that some positive power of every element is an idempotent.

Binary Operations : Equivalence Classes

Note. I don't how to make a letter with a line overtop of it so the equivalent notation here is *. ex) a* = a bar (a with a line overtop of it) Let M be a commutative monoid. Define a relation ~ on M by a ~ b if a = bu for some unit u. (a) Show that ~ is an equivalence on M and if a* deontes the equivalence class of a, let ...continues

A binary relation is defined on the set R of all real numbers. The problem is to prove that that binary relation is indeed an equivalence relation, and that there is a (well-defined) bijection between the set of equivalence classes and the set {x: x is a real number and 0 <= x < 1}.

Show that == (where == is the equivalence relation defined below) is an equivalence on A, and find a (well-defined) bijection %: A== -> B, where (a) A = R (the set of all real numbers) (b) B={x: x is an element of R and 0 <= x < 1} (c) for real numbers x and y, "x==y" (x is equivalent to y) if and only if x - y is an e ...continues

A (general) mapping % is defined on a (general) set A. Then a (general) mapping $ is defined on the set of equivalence classes for the kernel equivalence of %. The first part of the problem consists of proving that $ and % have certain properties. One of these is that % is the composite $*, where * is the natural mapping from A to the set of equivalence classes; another is that the cardinality of the set of equivalence classes is equal to the cardinality of %(A). In the second part, two specific pairs (A, %) are given, and the cardinality of the set of equivalence classes is to be found for each.

For a mapping %:A -> B, let == denote the kernel equivalence of %, and let *:A -> A== denote the natural mapping. Define $:A== -> B by $([a]) = %(a) for every equivalence class [a] in A==. 1. Show that $ is well defined and one-to-one, and that $ is onto if % is onto. Furthermore, show that % = $*, so that % is the composite ...continues

Binary Operations : Monoids

Let S be a set with an associative binary operation but with no identity. Choose an element 1 not belonging to S, write M = {1} or S, and define an operation on M by using the operation of S and 1s=s=s1 for all s belonging to S. Show that M is a monoid.