Consider the relation R = {(1,7), (3,3), (13,11)} and the relation S = {(1,1),(3

Question

Consider the relation R = {(1,7), (3,3), (13,11)} and the relation S = {(1,1),(3,11),(13,12),(15,1)} Identify dom(R), range(R),dom(S), range(S). The identity relation l_ A over a set A is defined by putting I = {(a, a), a belongs to A}. Identify dom(I_A) and ranged(I_A). Identify dom(A cross B). range(A cross B). Since relations are sets (of ordered pairs or tuples), we can apply to them all the concepts that we developed for sets, In particular, it makes sense to speak of one relation B being included in another relation S: every tuple that is an element of R is an element of 5. In this case, we also say that R is a subrelation of S. Likewise, it makes sense to speak of the empty relation: it is the relation that has no elements, and it is unique. It is thus the same as the empty set, and can be written

Explanation / Answer

Answer:

(a )

Dom ( R ) = ( 1, 3, 13 )

Dom ( S ) = ( 1, 3, 13, 15 )

Range ( R ) = ( 7, 3 11 )

Range ( S ) = ( 1, 11, 12, 1)

(b)

Functions - A function between A and B is a nonempty relation f A × B such that if (a, b) f and (a, b ) f , then b = b . The domain of f, denoted by dom f, is the set of all first elements of members of f. The range of f, denoted by ran f, is the set of all second elements of members of f. Symbolically

dom f = {a A : b B (a, b) f}

ran f = {b B : a A (a, b) f}

The set B is referred to as the codomain of f. If it happens that the domain of f is equal to all A, then we say that f is a function from A into B and we write f : A B.

If in the previous definition we allow the binary relation f to be such that (a, b) f , (a, b ) f and b = b , then we are talking of a different object, called a correspondence. When a function consists of just a few ordered pairs, it can be identified by listing them. But usually they are too many to list, so the function is defined by specifying the domain and giving a rule for determining the second element in the ordered pair that corresponds to a particular first element.

For example

f = (x, y) : y = x2 and x R

The domain of the function would be obtained either from the context or by stating it explicitly. In this part of the course we deal mainly with functions from R into R. Therefore,unless told otherwise, when a function is given by a formula, the domain is taken to be the largest subset of R for which the formula will always yield a real number.

( C) Since #(A) ¼ 2 and #(B) ¼ 3, #(AB) ¼ 2.3 ¼ 6. From the definition of a relation from A to B it follows that the set of all relations from A to B is just P(AB), and by a principle in the chapter on sets, #(P(AB)) ¼ 2#(AB) ¼ 26 ¼ 64.

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