What is an open interval?

1 Answer
Aug 31, 2016

Answer:

See explanation...

Explanation:

If #a# and #b# are Real numbers with #a < b# then #(a, b)# is used to denote the numbers which lie strictly between #a# and #b#. This is called "the open interval #a#, #b#".

In symbols we could write: #(a, b) = { x in RR : a < x < b }#

This reads #(a, b)# is the set of elements #x# in the set of Real numbers (#RR# for short) such that #a < x# and #x < b#.

When we want to talk about all the Real numbers we may write:

#(-oo, +oo)#

The symbols #-oo# (minus infinity) and #+oo# (plus infinity) are not really numbers. You can picture them as being at the extreme left and right ends of the Real number line. Any Real number #x# satisfies:

#-oo < x < +oo#

If we want to talk about any number greater than #5#, we can write:

#x in (5, +oo) " "# (#x# is in the open interval "#5# to plus infinity")

If we want to talk about any number less than #5#, we can write:

#x in (-oo, 5) " "# (#x# is in the open interval "minus infinity to #5#")

If #z != 5# then either #z < 5# so #z in (-oo, 5)# or #z > 5# so #x in (5, oo)#.

The amalgamation of these two sets is called the union and is denoted:

#(-oo, 5) uu (5, +oo)#