Let ω be a set of variables. we say that a propositional formula is a literal if the formula is equal to x or ¬x for x ∈ ω. we say that a propositional formula on ω is in conjunctive normal form if it is equal to =1 =1 ψi, j , where ψi, j is a literal. let φ be a propositional formula on ω. show using structural induction that there is a propositional formula ψ on ω in conjunctive normal form such that ψ |rho = φ |rho for any assignment rho to ω.

Which set of lengths could represent the sides of a triangle? a. 3, 7, 10 b. 4, 5, 10 c. 6, 8, 14 d. 12, 6, 7

Paula makes stained-glass windows and sells them to boutique stores. if her costs total $12,000 per year plus $4 per window for the frame. how many windows must she produce to earn a profit of at least $48,000 in one year if she sells the windows for $28 each? 1. define a variable for the situation. 2. write an inequality that represents her profit. note: revenue is money coming in. cost is money going out. profit is the difference between the revenue and the cost. in other words: revenue - costs profit 3.using words, describe how many windows she must sell to have a profit of at least $48,000.

Arandom sample of n = 25 observations is taken from a n(µ, σ ) population. a 95% confidence interval for µ was calculated to be (42.16, 57.84). the researcher feels that this interval is too wide. you want to reduce the interval to a width at most 12 units. a) for a confidence level of 95%, calculate the smallest sample size needed. b) for a sample size fixed at n = 25, calculate the largest confidence level 100(1 − α)% needed.