Answer to Question #91152 in Discrete Mathematics for GATARE james

Question

a) A market researcher investigating consumers preference for three brands of beverages namely; coffee, tea and cocoa in Thika town gathered the following information.
From a sample of 800 consumers, 230 took coffee, 245 took tea and 325 took cocoa, 30 took all the three beverages, 70 took coffee and cocoa, 110 took coffee only, 185 took cocoa only.
Required:
i). Present the above information in a venn diagram. [2mks]
ii). The number of consumers who took tea only. [2mks]
iii). The number of customers who took tea and cocoa only. [2mks]
iv). The number of customers who took tea and cocoa only. [2mks]
v). The number of customers who took none of the beverages. [2mks]

Answer 1

Let set A = consumers who took coffee

set B = consumers who took tea

set C = consumers who took cocoa

then

|A| = 230

|B| = 245

|C| = 325

|AC| = 70

|(A/B)/C| = 110

|(C/A)/B| = 185

|ABC| = 30

Two useful formulas:

EF + E/F = EF + EF^c = E(F + F^c) = E, where F^c is complement set of F

EF + E/F = E (1)

using (1)

XZ + XY/Z + X/Y/Z = (XY + X/Y)Z + XY/Z + X/Y/Z = (XYZ +XY/Z) + (X/Y)Z +X/Y/Z =

XY +X/Y = X :

XZ + XY/Z + X/Y/Z = X (2)

using (1) again

AC/B + ABC = AC/B + ACB = AC: |AC/B| = |AC| - |ABC| = 70 - 30 =40 number of customers who took coffee and cocoa only

using (2): AC + AB/C + A/B/C = A: |AC| + |AB/C| + |A/B/C| = |A|,

|AB/C| = |A| - |AC| - |A/B/C| = 230 - 70 - 110 = 50 number of customers who took coffee and tea only

analogously:

|BC/A| = |C| - |C/A/B| - |AC| = 325 - 185 - 70 = 70 number of customers who took tea and cocoa only

using (1)

|AB| = |AB/C| + |ABC| = 50 + 30 = 80 number of customers who took coffee and tea

using (2)

|B/A/C| = |B| - |AB| - |BC/A| = 245 - 80 - 70 = 95 number of customers who took tea only

Number of customers who took tea coffee or cocoa:

|A/B/C| + |B/C/A| +|C/A/B| + |AB/C| + |AC/B| + |BC/A| + |ABC| =

110 + 95 + 185 + 50 + 40 + 70 + 30 = 580

Number of customers who took none of the beverages is 800 - 580 = 220

Answer: the number of customers who took tea only is 95,

the number of customers who took tea and cocoa only is 70,

the number of customers who took tea and coffee only is 50,

the number of customers who took none of the beverages is 220.