# Discrete Mathematics And Its Applications 6th Edition By Kenneth H. Rosen

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rosen, discrete mathematics and its applications, 6th edition

The following statement is a conditional proposition in one of its many alternate forms. Write it in English in the form “If then .” “To pass the course it is necessary that you get a high grade on the ﬁnal exam.” Solution: We ﬁrst rephrase the statement as “Getting a high grade on the ﬁnal exam is a necessary condition for passing the course.” The word “necessary” and the word “suﬃcient” give rise to converse implications: “p is suﬃcient for q” is p → q, while “p is necessary for q” is q → p (or, .rosen, discrete mathematics and its applications, 6th edition

The following statement is a conditional proposition in one of its many alternate forms. Write it in English in the form “If then .” “To pass the course it is necessary that you get a high grade on the ﬁnal exam.” Solution: We ﬁrst rephrase the statement as “Getting a high grade on the ﬁnal exam is a necessary condition for passing the course.” The word “necessary” and the word “suﬃcient” give rise to converse implications: “p is suﬃcient for q” is p → q, while “p is necessary for q” is q → p (or, .rosen, discrete mathematics and its applications, 6th edition

Solution: The solution depends on what we take for the universe for the variable. If we take all Juniors in this class as the universe, we can write the proposition as ∀x S(x) where S(x) is the predicate “x scored above 90 on the ﬁrst exam.” However, if we take all students in this class as the universe, then we can write the proposition as ∀x (J(x) → S(x)) where J(x) is the predicate “x is a Junior.” We can extend the universe still further. Suppose we take all students as the universe. Then we need to .rosen, discrete mathematics and its applications, 6th edition

Solution: The solution depends on what we take for the universe for the variable. If we take all Juniors in this class as the universe, we can write the proposition as ∀x S(x) where S(x) is the predicate “x scored above 90 on the ﬁrst exam.” However, if we take all students in this class as the universe, then we can write the proposition as ∀x (J(x) → S(x)) where J(x) is the predicate “x is a Junior.” We can extend the universe still further. Suppose we take all students as the universe. Then we need to .rosen, discrete mathematics and its applications, 6th edition

Prove that there is only one pair of positive integers that is a solution to 3x2 + 2y 2 = 30. Solution: The two terms on the left side of the equation each involve a square, and hence the sum exceeds 30 for small values of x andIn particular, in order to have a solution, we must have x ≤ 3 and y ≤The number of such pairs is suﬃciently small that we can use an exhaustive proof, considering the cases (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4). The only .**Suggested**

discrete mathematics and its application 6th edition by kenneth h. rosen downlaod

discrete mathematics and its application 6th edition by kenneth h. rosen

solutions rosen, discrete mathematics and its applications, 6th edition extra examples section1.3

student's solutions guide to accompany discrete mathematics and its applications, 6th edition

student\'s solutions guide to accompany discrete mathematics and its applications 6th edition

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