MTH104 Final Term Slides Lectures 23-45 by Mahar Afaq

P rove that the set R of real numbers has the power of continuum. P roof.

Each set A is assigned a symbol in such a way that two sets A and B are assigned the same symbol if and only if they are equipotent. This symbol is called the cardinality or cardinal number of A, and it is denoted by |A|, n (A) or card(A) Thus,

0 is assigned to the empty set ø, and n is assigned to the set {1, 2, … n }. Thus,

Alternatively, the symbols 0, 1, 2, 3,… are assigned, respectively, to the sets Cardinal numbers of infinite sets are called infinite or transfinite cardinal numbers. The cardinal number of the infinite set P of positive integers is Thus,

In particular, The cardinal number of the unit interval I = [0, 1] is denoted by c and it is called the power of continuum. Thus:

In particular, we have |R | = c, and the cardinal number of any interval is c. ( a A is denumerable or countably infinite means |A| = ( b A is countable means |A| is finite or |A| = ( c A has the power of continuum if |A| = c.

Let A and be two sets. We say that |A| ≤|B | if there exists a one-to-one(injective) function f : A → B . The above relation between cardinal numbers are well defined, that is, the relations are independent of the particular sets involved Namely if particular sets involved. Namely, if

Example 1

Let A be a proper subset of a finite set B .

Example 2

Let n be a finite cardinal.

We may write

Example 3

Consider the set P of positive integers and the unit interval I, that is, consider the sets

Th f i f P I d fi d b f( 1/ i The function f : P I defined by f( n ) = 1/ n is one-toone.(prove it.) Prove that

Addition and multiplication of the counting numbers N are sometimes treated from the point of view of set theory. The interpretation of 2+3 = 5, for example, is the union of two disjoint sets, one having two elements and the other having three elements is a set with five and the other having three elements, is a set with five elements. This idea leads to a completely general definition of addition of cardinal numbers. Definition: Let α and β be cardinal numbers and let A and B be disjoint sets with α = |A| and β = |B|. Then the sum of α and β is denoted and defined by α + β = |(A ∪B)|

Note:(1) This definition is well-defined.

(2) If A and B are any two sets, then