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Please show work if needed!Discrete Structures: Homework #1 [2 points] Given a year X as a natural number, John only considers X as a leap year when (and only when) X fits into any one of the following descriptions 1. (i) X is divisible by 4 and X is divisible by 400 (ii) Xis divisible by 4 and X is not divisible by 100 Let D4, D100 and D400 denote the atomic propositions regarding whether X is divisible by 4, by 100, and by 400 respectively. Please translate Johns way of determining whether X is a leap year into a compound logic proposition involving the atomic propositions above, parentheses if needed, and the logic operators л, v, and . 2. [2 points] Consider the compound proposition you have for problem#1 above. Is the compound proposition true when X is 1776? How about when X is 1800, 1945, or 2000 respectively? 3. [2 points] Given a year X as a natural number, Mary considers X as a leap year when (and only when) X fits into none of the following descriptions (i) (ii) Xis not divisible by 4, Xis divisible by 100 and Xis not divisible by 400 Let D4, D100 and D400 denote the atomic propositions regarding whether X is divisible by 4, by 100, and by 400 respectively. Please translate Marys way of determining whether X is a leap year into a compound logic proposition involving the atomic propositions above, parentheses if needed, and the logic operators ^, v, and- . [2 points] Consider the compound proposition you have for problem#3 above. Is the compound osition true when X is 1776? How about when X is 1800, 1945, or 2000 respectively? 4, prop [2 points] Lets refer to the compound proposition you got in Problem #1 as Johns proposition J and refer to the compound proposition you got in Problem #3 as Marys proposition M Create a truth table to check whether J and M are logically equivalent in all situations. Are they equivalent based on your finding? Why or why not? 5.

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