Using the principle of mathematical induction prove that 5 10 15+...+ 5n= 5n(n

Let A and B be the nxn matrices and I be the identity matrix of order n x n. Check the validity of the following statements and give justification:

<img src='./qimages/12281-1b.jpg'>

Let B^2 B be a function defined as and 1. Find the Boolean expression specifying the function f.

Let f be a permutation function defined as follows:

Find the inverse of f i.e.,

Make a table to recursively calculate Pn^k where n is the total number, k is the number of partitions, using the following conditions

7 n 1 and 1 k 7.

An urn contains 15 balls, of which eight are red and seven are black. In how many ways can 5 balls be chosen such that two are red and three are black?

In how many ways can 7 people be seated around a circular table?

Show that p is a tautology.

Prove:

<img src='./qimages/12281-2b.jpg'>

Give the direct proof of the statement "The sum of two odd integers is always even".

Explain the Identity Laws of Boolean algebra.

Reduce the following Boolean expressions to simpler form:

<img src='./qimages/12281-3a.jpg'>

Find the Boolean expression for the following circuit

<img src='./qimages/12281-3b.jpg'>

Make the circuit corresponding to the following Boolean expression

<img src='./qimages/12281-3c.jpg'>

Write the set expressions for the following Venn diagrams:

<img src='./qimages/12281-4a.jpg'>

What is an equivalence relation? Let A be a set and R be an equivalence relation on A such that A/R Write R.

Let f and g be the two functions such that x2 and 2x. Define fof, fog, gof and gog.

Find the number of distinguishable words that can be framed from the letters of 'MISSISSIPPI'.

Prove:

<img src='./qimages/12281-5a.jpg'>

Use pigeonhole principle to show that if 7 colours are used to paint 50 bicycles, then at least 8 bicycles will have the same colour.

In how many ways can 10 students be grouped into 2 groups?

Obtain the truth value of the disjunction of 'Sun moves around the Earth' and