on page 88; section 2.5 there is a blue box that has cardinal numbers of infinite sets, where it has rael numbers as having the cardinal number of c(continuum. what all is included in the set of real numbers?
The real numbers consist of all integers, all rational numbers of form p / q, and all irrational numbers.
Rational numbers have decimal expansions that either terminate or repeat. For example 1/3 = .333..., while 1/4 = .25, and 3/7 = .48571... . The rational numbers can be put into 1-1 correspondence with the whole numbers, even though they lie 'densly' on the real line (no matter how close you want to get to a rational number, there is another rational number closer than that). It would seem that there is no space left since the rational numbers are that densely distributed on the number line, but there are.
The irrational numbers have decimal expansions that neither repeat nor terminate. Probably the most famous example is pi = 3.1415926 etc. , which not only fails to repeat or terminate but has no perceivable pattern whatsoever. Irrational numbers can also have patterns, like 1.01001000100001... , which has a predictable pattern but does not repeat or terminateand is hence not a rational number. Without the restriction of repeating or terminating there are a lot more possibilities than there are with these restrictions, so that the irrational numbers cannot be put into 1-1 corresondence with the whole numbers.
The real numbers are not just dense on the number line, they completely cover it, in the sense that every point of the number line is represented by a real number. We therefore say that the real numbers are continuous on the real line, hence the term 'continuum'.
The cardinality of the set of real numbers is therefore greater than aleph-null. We call this cardinality c.
when dealing with a one-to-one correspondanceand using a variable to describe the pattern. exa.1,2,3,4,5...as n and 2,4,6,8,10 as 2n. how could a set with the multiples of 3 and 5 be set up and defined in a one-to-one correspondance with a set of real numbers?
These sets are not in 1-1 correspondence with the set of real numbers, but they are in one-to-one correspondence with the set of whole numbers, and hence with one another.
Every positive multiple of 3 is of the form 3 n, with n a whole number.
Every positive multiple of 5 is of the form 5 n, with n a whole number.
So a 1-1 correspondence between these sets would be indicated by 3n <-> 5n.