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write the following sentence in predicate calculus.
Every teacher takes care of his student.john is the teacher and James is his student.

How many solutions are there to the equation
x1+x2+x3+x4+x5=21,
where xi, i=1, 2, 3, 4, 5, is a non negative integer such that
b) xi≥1 for i=1, 2, 3, 4, 5?
c) 0≤x1≤11?

A plane sheet of material is bound by the curve y = x^2 from x = 0 to x =1, the x-axis
and the line x =1. If the mass per unit area (density) of the sheet is xy find the mass of
the sheet.

A trade development board organized coventions abroad for its 150 members in the jewelers industry. The coventions fell under three categories trading, manufacturing and design 52 attended the trading category, 46 the manufacturing and 32 the design; 20 attended the trading and manufacturing category and 18 the trading and design; 8 attended the manufacturing and design but not the trading category and 14 attended the trading and manufacturing but not the design category. draw a venn diagram to illlustrate the above information and find the number of members who attended
i. none of the convention
ii. at least two categories of the coventions
iii. akk the three categories of the conventions.

What is the number of perfect matching of a complete graph Kn with n vertices?

Find the transitive closure of R = {(a, a), (b, a), (b, c), (c, a), (c, c), (c, d), (d, a), (d, c)} on the set {a, b, c, d}.

a)Consider the decimal number a = 137.
i.Find in set builder notation the set of all positive integers b such that b ≡ a (mod 5).
ii.Is the number a prime? Explain.
iii.Convert the number a to binary and octal numbers

Let R be the partial order relation defined on A = {2, 3, 4, 5, 6, 8, 10, 40}, where xRy means x | y.
i.Draw the Hasse diagram for R.
ii.Find the upper and lower bounds of {4, 8}.

Find the smallest equivalence relation on {1, 2, 3} that contains (1, 2) and (2, 3).

How many solutions are there to the equation
x1+x2+x3+x4+x5=21,
where xi, i=1,2,3,4,5, is a nonnegative integer such that
a)x1>=1?
b)xi>=2, for i=1,2,3,4,5?
c)0<=x1<=10?
d)0<=x1<=3, 1<=x2<4, and x3>=15?

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