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Prove or disprove the following: 

1. lg√n ∈ O(lg n) (√n means square root of n)

2. lg n ∈ O(lg√n) 

3. 2n+1 ∈ O(2n)

Prove the following using mathematical induction:
1. (ab)n = an bn for every natural number n
2. 13 +23+33+…+n3 = (1+2+3+…+n)2
3. 1+3+5+7+…+(2n-1) = n2
Prove that n + log2n = O(n) by showing that there exists a constant c > 0 such that n + log2n ≤ cn.
(note that log2n means (log n)2.)
1. Prove that n + log n = O(n) by showing that there exists a constant c > 0 such that n + log n ≤ cn.
1. Write a complete Main method that prints Hello, world to the screen.

2. Suppose your name was George Gershwin. Write a complete main method that would print your last name, followed by a comma, followed by a space and your first name.
3. Write an expression that attempts to read a double value from the TextBox textBox and stores it in a Double variable, x, that has already been declared.
Write a program to output the following:

**********
* *
* PYTHON *
* *
**********
Write a program to output the following:

**********
* *
* PYTHON *
* *
**********
Write a program that uses two input statements to get two words as input. Then, print the words on one line separated by a space.
Write a function matched(s) that takes as input a string s and checks if the brackets "(" and ")" in s are matched: that is, every "(" has a matching ")" after it and every ")" has a matching "(" before it. Your function should ignore all other symbols that appear in s. Your function should return True if s has matched brackets and False if it does not.

Hint: Keep track of the nesting depth of brackets. Initially the depth is 0. The depth increases with each opening bracket and decreases with each closing bracket. What are the constraints on the value of the nesting depth for all brackets to be matched?

Here are some examples to show how your function should work.


>>> matched("zb%78")
True

>>> matched("(7)(a")
False

>>> matched("a)*(?")
False

>>> matched("((jkl)78(A)&l(8(dd(FJI:),):)?)")
True
Write a program that computes the results of integer and floating-point division, where variables a, and b are declared as integers, c and d are declared as floating-point numbers, and all are user-specified.

Integer division is defined as whole-number division. For example, 1/2 (as integers) = 0. Whereas, if we specified 1/2 (as "floats"; i.e., floating-point numbers), the result is 0.5. In C++, the result of an arithmetic operation (e.g., whether the computer computes the result as an integer or as a float) is specified by the data type.
Your task is to make edits within main such that, when run, the program produces the following console output:
Enter two integers: [user-specified value for a] [user-specified value for b]
Integer division: [a]/[b] = [a]/[b]
Floating-point division: [c]/[d] = [c]/[d]
Mixed-type division: [a]/[d] = [a]/[d]
Mixed-type division: [c]/[b] = [c]/[b]
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