The recursive definition given below defines a set S of strings over the alphabet {a, b}: Base case: λ ∈ S and a ∈ S Recursive rule: if x ∈ S then, xb ∈ S (Rule 1) xba ∈ S (Rule 2) This problem asks you to prove that the set S is exactly the set of strings over {a, b} which do not contain two or more consecutive a's. In other words, you will prove that x ∈ S if and only if x does not contain two consecutive a's. The two directions of the "if and only if" are proven separately. (a) Use structural induction to prove that if a string x ∈ S, then x does not have two or more consecutive a's.

1- is the product of two rational numbers irrational or rational? first, make a hypothesis by multiplying two rational numbers. then, use variables such as x=a/b and y=c/d and the closure property of integers to prove your hypothesis. 2- what do you think the product of a nonzero rational number and an irrational number is? is it rational or irrational? make use of variables, the closure property of integers, and possibly a proof by contradiction to prove your hypothesis. 3- why do we have to specify that the rational number must be nonzero when we determine what the product of a nonzero rational number and an irrational number is? if the rational number were 0, would it give us the same result we found in part b?

What is the distance from zero if a quadratic function has a line of symmetry at x=-3 and a zero at 4

Parabolas y=−2x^2 and y=2x^2+k intersect at points a and b that are in the third and the fourth quadrants respectively. find k if length of the segment ab is 5.