Cambridge Additional Mathematics

16 Sets and Venn diagrams (Chapter 1)

CLOSED AND OPEN INTERVALS

Anintervalis a connected subset of the number lineR.

An interval isclosedifbothof its endpoints are included.

An interval isopenifbothof its endpoints arenotincluded.

INTERVALS WHICH OVERLAP

When two intervals overlap, we consolidate them into a single interval.

For example: [¡ 2 ,5)[[1,7) = [¡ 2 ,7)

EXERCISE 1B

1 Explain whether the following sets are finite or infinite:

a fx 2 Z:¡ 26 x 61 g b fx 2 R :¡ 26 x 61 g c fx 2 Z:x> 5 g

d fx 2 Q :0 6 x 61 g e (2,4) f [¡ 3 ,7]

g (¡1,0)

2 For the following sets:

i Write down the meaning of the interval notation.

ii If possible, list the elements ofA.

iii Find n(A).

In this course

02 =N.

-2 1 5 7 x

iv If possible, sketchAon a number line.

a A=fx 2 Z:¡ 16 x 67 g b A=fx 2 N :¡ 2 <x< 8 g

c A=fx 2 R :0 6 x 61 g d A=fx 2 Q :5 6 x 66 g

e A=[¡ 1 ,5) f A=fx 2 R :3<x 65 [x> 7 g

g A=(¡1,1][(2, 1 ) h A=(¡1,2)[[1, 1 )

3 Write in interval notation:

a the set of all integers between¡ 100 and 100

b the set of all real numbers greater than 1000

c the set of all rational numbers between 2 and 3 , including 2 and 3.

For x 2 R, we commonly use the following notation to concisely write intervals:

[a,b] represents the closed interval fx 2 R : a 6 x 6 bg

[a,b) represents the interval fx 2 R : a 6 x<bg

(a,b] represents the interval fx 2 R : a<x 6 bg

(a,b) represents the open interval fx 2 R : a<x<bg

An interval which extends to infinity has no defined endpoint.

So, for fx 2 R :x>ag we write [a, 1 ).

This shorter notation is not

needed for the syllabus.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_01\016CamAdd_01.cdr Tuesday, 8 April 2014 10:20:43 AM BRIAN