1. Atwo-pole 60 Hz induction motor has a slip of2%on full load. The full-load speed is (a) 58.8 r/s (b) 3528 r/min (c) 60 r/s (d) 2880 r/min 2.The….

## Prove that the hypothesis class of all conjunctions over d variables is PAC learnable and bound its sample complexity.

1. In this question, we study the hypothesis class of *Boolean conjunctions *defined as follows. The instance space is *X *={0*,*1}*d *and the label set is *Y *={0*,*1}. A literal over the variables *x*1*, . . ., **xd *is a simple Boolean function that takes the form *f *(x)= *xi*, for some *i *∈ [*d*], or *f *(x) = 1−*xi *for some *i *∈ [*d*]. We use the notation .*xi *as a shorthand

for 1− *xi *. A conjunction is any product of literals. In Boolean logic, the product is denoted using the ∧ sign. For example, the function *h*(x) = *x*1 · (1− *x*2) is written as *x*1 ∧ .*x*2. We consider the hypothesis class of all conjunctions of literals over the *d *variables. The empty conjunction is interpreted as the all-positive hypothesis (namely, the function that returns *h*(x) = 1 for all x). The conjunction *x*1 ∧ .*x*1 (and similarly any conjunction involving a literal and its negation) is allowed and interpreted as the all-negative hypothesis (namely, the conjunction that returns *h*(x) = 0 for all x). We assume realizability: Namely, we assume that there exists a Boolean conjunction that generates the labels. Thus, each example (x*, **y*) ∈ *X *× *Y *consists of an assignment to the *d *Boolean variables *x*1*, . . ., **xd *, and its truth value (0 for false and 1 for true). For instance, let *d *= 3 and suppose that the true conjunction is *x*1 ∧ .*x*2. Then, the training set *S *might contain the following instances: ((1*,*1*, *1)*, *0)*, *((1*,*0*, *1)*, *1)*, *((0*,*1*, *0)*,*0)((1*,*0*, *0)*, *1).

Prove that the hypothesis class of all conjunctions over *d *variables is PAC learnable and bound its sample complexity. Propose an algorithm that implements the ERM rule, whose runtime is polynomial in *d *·*m*.