QUESTION ONE

a$)$ Answer statements i to vii as either TRUE or FALSE and the rest appropriately.

$1.$ The intersection of sets A and B is the set of all elements in A and B$.$

$2.$ A $$ B $=($AUB$)-($A $\backslash$cap B$)$

$3.$ All sets have at least two subsets.

$4.$ The union of any set and its complement results in the universal set.

$5.$ The set $\{5,10,15,20............\}$ is finite set.

$6.$ The compliment of A relative to B is the set of elements in B that are not in A$.7.$ If set $($A $\backslash$cap B$)=\{\},$ then sets A and B are disjoint

$8.$ Every set has at least one subset.

$9.\{$a$\}$ is an element of $\{$a$,$ b$\}.(9$ marks$)$

b$)$ Assume we have the following sets. Let U$=\{1,2,3,4,5,6,7,8,9,10\},$ P $=\{2,5,7,9\},$ Q $=\{1,3,5,6,9\}$ and R $=\{2,4,6,8,10\}.$ Get the following sets:

i$)$ P $$ Q ii$)$ P $$ R iii$)$ Q $$ R iv$)($Q $$ R$)$

v$)$ Q$\text{'}$ R$\text{'}$ vi$)$ Q $$ R$\text{'}$ vii$)$ P $$ Q $$ R viii$)$ P $-($Q $$ R$)$

ix$)$ Q$$ R $(9$ marks$)$

c$)$ Research by Umoja Consultants of the characteristics of $300$ enterprises which had failed revealed that $285$ of them either had poor finance, poor management or a poor product. $12$ of the enterprises had all three of these characteristics. $120$ enterprises were poorly financed but had good management and good product. $21$ were poorly financed and had poor management. $27$ were poorly financed and had poor product. $45$ enterprises had poor management but sufficient finance and good product. $30$ enterprises had poor management and poor product.

Required

i$)$ Represent the above data using a Venn diagram $(6$ marks$)$

ii$)$ How many businesses failed due to only one characteristic?

$(3$ marks$)$

iii$)$ Determine the most prevalent characteristic for business failure

$(3$ marks$)$