Define the following propositions: j: Sally got the job. l: Sally was late for her interview r: Sally updated her resume. Express each pair of sentences using logical expressions. Then prove whether the two expressions are logically equivalent. (a) If Sally did not get the job, then she was late for her interview or did not update her resume. If Sally updated her resume and did not get the job, then she was late for her interview. (b) If Sally did not get the job, then she was late for her interview or did not update her resume. If Sally updated her resume and was not late for her interview, then she got the job. (c) If Sally got the job then she was not late for her interview. If Sally did not get the job, then she was late for her interview. (d) If Sally updated her resume or she was not late for her interview, then she got the job. If Sally got the job, then she updated her resume and was not late for her interview.

John and daphne are saving for their daughter ellen's college education. ellen just turned 10 at (t = 0), and she will be entering college 8 years from now (at t = 8). college tuition and expenses at state u. are currently $14,500 a year, but they are expected to increase at a rate of 3.5% a year. ellen should graduate in 4 years--if she takes longer or wants to go to graduate school, she will be on her own. tuition and other costs will be due at the beginning of each school year (at t = 8, 9, 10, and 11).so far, john and daphne have accumulated $15,000 in their college savings account (at t = 0). their long-run financial plan is to add an additional $5,000 in each of the next 4 years (at t = 1, 2, 3, and 4). then they plan to make 3 equal annual contributions in each of the following years, t = 5, 6, and 7. they expect their investment account to earn 9%. how large must the annual payments at t = 5, 6, and 7 be to cover ellen's anticipated college costs? a. $1,965.21b. $2,068.64c. $2,177.51d. $2,292.12e. $2,412.76