How many bitstrings of length 10 are there that contain 5(or more) consecutive 0’s or contain 5(or more) consecutive 1’s?

How many bitstrings of length 10 are there that contain 5(or more) consecutive 0’s or contain 5(or more) consecutive 1’s? Justify your answer.

Prove that S is finite

Let R ba a partial order on S, and suppose that x is a unique minimal element in S. a) prove that S is finite, then xRy for all s in S b) show that the conclusion in (a) need not be true is S is infinite

Symmetry and recursive definitions

1) Find a relation R on a set S that is neither symmetric nor antisymmetric 2) Let S be a set containing exactly n elements. How many antisymmetric relations on S are there. 3) give a recursive definition of X^n for any positive integer n 4) give a recursive definition of the nth odd positive integer 5) Let g: Z -> Z ...continues

Mathematical induction and geometric progression

Please see attached sheet for questions about induction, geometric progression, interest, and recurrence relations.

Boolean expressions and circuits

Please see attached file .

Discrete structures in mathematics and computer science

Q1) Use the standard logical equivalences to simplify the expression (ㄱp ^ q) v ㄱ(pVq) Q2) consider the following theorem "The square of every odd natural number is again an odd number" What is the hypothesis of the theorem? what is the conclusion? give a direct proof of the theorem. Q3) consider the follo ...continues

Sets

Qu1) Is it true that ρ(A∪B)= ρ(A) ∪ ρ(B)? justify your answer. Qu2) Consider the function f:A→A defined by f(x)=x+1 and justify your answers. a) For A=Ν (integers) is f onto? b) For A=R(real number) is f injective? c) For A=Q (rationals) is f onto? d) For A=Z(all integers) is f a bije ...continues

Prove using induction

Prove using induction--- 1+α+α2+.......+αn-1=αn-1/a-1

Recurrence relation problem.

I need to know how to solve this problem: Solve the following recurrence relation: x(n) = 3x(n-1) for n > 1, x(1) = 4. It requires backwards substitution to solve.

How many one-to-one functions are there between A and B?

If A has eight unique elements and B has eight unique elements, how many one-to-one functions would there be? Also, I'd like to know how many there would be if A happened to have less elements than B or if B had less elements than A. Thank you for your time.