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According to information obtained from mathematics department regarding three mathematics units done by 100 students, those who are doing calculus are 45, those doing discrete are 49 and those doing statistics are 38. Those doing calculus and discrete are 17, those doing calculus and statistics are 12 and those doing the three units are 5. Use Venn diagram to find the number of students doing discrete and statistics but not calculus
What is the value of k after the following code has been executed?
k := 0
for i1 := 1 to n1
for i2 := 1 to n2
for im := 1 to nm
k := k + 1
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.
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]
Show that the following argument form is valid.

p --> q
q --> r

∴ p --> r
How many 20 digit binary numbers have four 1's in them?
draw a graph that has the following vertices and edges.
V = {x1, x2, x3, x4} and E = {{x1, x3, }, {x1, x4}, {x2, x4}}
draw the graphs that have the following vertices and edges
V = {a, b, c}, and E = {{a, b}, {b, c}}.
1. Expand in a two-element universe (the elements are named 'a' and 'b')

(a) ~(x) ((Fx v Gy) v Ka)

(b) (x) ~ (Kx v Ka)

(c) (Ex) (Cy v (Fx --> ~Ga))

2. For the following wffs, indicate which variables are free and which are bound (you can use 'F' for free and 'B' for bound.) Make sure you draw a vertical line underneath each variable with the letters 'F' or 'B' at the bottom of each vertical line.

You can write the wffs with the vertical lines on a piece of paper and submit a photo of it.

(a) (x) (y) (z) ((Fxy --> Gxy) v (Hxa --> Hzbu))

(b) (Ex) (Ey) Hxxy --> (z) (Gz v Hx)

(c) (z) (Ex) (y) (Axayzw v Bxycuvz)
The contrapositive of 'there exist y€z such that p(y) is true' is ' there exist x€z such that p(x) is true'.
Is the statement true or false? Justify your answer.
A school bus is transporting athletes home from an athletics meeting. Of these athletes 15% took part in hurdles,20% took part in the long jump event 35%,took part in relay races,30% ran the 100-m sprint and 25% took part in the high-jump event. Explain how is this possible.
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