6. A coin with Heads probability p is tossed repeatedly. What is the expected number of tosses needed to get k successive heads?

The domain of discourse for this problem is a group of three people who are working on a project. To make notation easier, the people are numbered 1, 2, 3. The predicate M(x, y) indicates whether x has sent an email to y, so M(2, 3) is read "Person 2 has sent an email to person 3." The table below shows the value of the predicate M(x, y) for each (x, y) pair. The truth value in row r and column y gives the truth value for M(x, y). M | 1 2 | 3 1 TTT 2 TFT 3 TT|F Determine if the quantified statement is true or false. Justify your an- swer. (a) Væ Vy (x # y) → M(x, y)) (b) Væ3y ¬M(x, y) (c) 3r Vy M(x, y)

John Wallis and other 17th century mathematicians were struggling with how to fit negative numbers into arithmetic without introducing inconsistencies. Explain the reasoning leading to Wallis's paradoxical claim that, if negative numbers are less than zero, they must be greater than infinity. Wallis's argument depends, in part, on the assertion that (3/-1) = -3. Must this be true? Why or why not? If a student proposed Wallis's argument, how would you respond?

The algebraic translation of "x is no less than 7" is .

Can you answer subpart d now that I am using a new question

The notation A*B*C is shorthand for “the point B is between point A and point C.” Can I please see the correct proof for the problem taking this notation into account?

is there a shorter way to do this where you don't get decimals in your answer?