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A. Answer the following questions: B. Q1. Write down the following set in rule method: C = {2,5,10,17,26,37,50} Q2. Construct a truth table for the proposition: (pq)v(~p-~q) Q3. Show that the composition a * b = ab,a, b E R is not associative. Q4. Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination. Answer the following questions: Q1.Show that the relation R 'less than or equal to 'on the set of integers is a partial order relation. Q3. Find out the units of Z8 = {0,1,2,3,4,5,6,7} the ring of integers modulo 8. Q4. Prove that (PQ) (~P VQ) Q2. Examine whether R is a group under the operation * defined by a * b = 2(a + b) Va, b E R. C. Answer the following questions: (i) (ii) (iii) 2 X 4 =8 AA is symmetric matrix. A + A' is symmetric and A- A/is skew symmetric A is the sum of a symmetric and a skew symmetric matrix. Q1. How many combinations and permutations can be made with the letters of the word "SCIENCE" taken three at a time? Q2. If A be a square matrix, show that D. Answer the following questions: 3 X 4 = 12 5 X 2 = 10 a) x + y - z=1 2x + 3y + z = 3 x + 2y +3z = 2 Q1. Let F be the set of all functions f: R R. For f, g EF,let us define f + g and fg as follows: (f + g)x= f(x) + g(x) Vx R (b) x + 2y + z = 1 2x + 3y - 2z = 2 x +y + z = 3 10 X 2 = 20 (fg)x= f(x) g(x)=x ER Show that F is a commutative ring with unity under the above defined operations. Q2. Determine the values of so that the following system has(i) a unique solution (ii) no solution (iii) infinite number of solutions.