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15. Find a useful denial (i.e., a statement logically equivalent to the negation) of each statement, and express it in mathematically precise, natural English. Express all conditional statements in the form "if ..., then ...". Do not add implied quantifiers. (Here, a, b, c, n are fixed integers, f is a fixed function, and x0, L are fixed real numbers.) (a) n is not a multiple of 4 if n is even. (b) If f has a relative maximum at x0 and f is differentiable at x0, then f'(x0) = 0. (c) For every integer m, m^2 is odd and m^3 - 1 is divisible by 4. (d) For all integers j and k, if n = jk, then j = 1 or k = 1. (e) If n is a perfect square, then there exists an integer k such that n = 3k or n = 3k + 1. (f) bc is divisible by a only if b is divisible by a or c is divisible by a. 

          15. Find a useful denial (i.e., a statement logically equivalent to the negation) of each statement, and express it in mathematically precise, natural English. Express all conditional statements in the form "if ..., then ...". Do not add implied quantifiers. (Here, a, b, c, n are fixed integers, f is a fixed function, and x0, L are fixed real numbers.)
(a) n is not a multiple of 4 if n is even.
(b) If f has a relative maximum at x0 and f is differentiable at x0, then f'(x0) = 0.
(c) For every integer m, m^2 is odd and m^3 - 1 is divisible by 4.
(d) For all integers j and k, if n = jk, then j = 1 or k = 1.
(e) If n is a perfect square, then there exists an integer k such that n = 3k or n = 3k + 1.
(f) bc is divisible by a only if b is divisible by a or c is divisible by a.
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15. Find a useful denial (i.e., a statement logically equivalent to the negation) of each statement, and express it in mathematically precise, natural English. Express all conditional statements in the form "if ..., then ...". Do not add implied quantifiers. (Here, a, b, c, n are fixed integers, f is a fixed function, and x0, L are fixed real numbers.) (a) n is not a multiple of 4 if n is even. (b) If f has a relative maximum at x0 and f is differentiable at x0, then f'(x0) = 0. (c) For every integer m, m^2 is odd and m^3 - 1 is divisible by 4. (d) For all integers j and k, if n = jk, then j = 1 or k = 1. (e) If n is a perfect square, then there exists an integer k such that n = 3k or n = 3k + 1. (f) bc is divisible by a only if b is divisible by a or c is divisible by a.

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Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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Find the useful denial (i.e., a statement logically equivalent to the negation) of each statement and express it in mathematically precise, natural English. Express all conditional statements in the form "if...then." Do not add implied quantifiers. (Here, n, f, and L are fixed integers, a fixed function, and fixed real numbers, respectively.) 1. n1 is a multiple of 4 if n is even. 2. If f has relative HAXIT at fo and f is differentiable at To, then f'(xo)
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Transcript

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00:01 So from here what we understand from negation of this statement like negation of statement a or b is not a and not b.
00:17 Next negation of statement a and b is that either not a or a or not b.
00:38 And if a statement is in the form of if a then b, then for that statement to be that false, b should be false when a holds.
01:02 And if a holds, then b is false.
01:04 So, negation is a and not b.
01:08 So, from that so we have to keep in mind all this term while solving this question...
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