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3. We define the absolute value function by: for all î ¤ R, we have { |x| := X -X if x ≥ 0 if x < 0 Prove that for all x ER, we have x² ≥ 0. 4. Let x be a real number. Prove that if x² = x, then x < 2. 5. Prove that if n is an integer, then n² + 3n+1 is odd. 6. Let a, ,b be integers. Prove that if a + b is even, then a - b is even. 7. Let a, b be integers. Prove that if ab is even, then a or b is even.

3. We define the absolute value function by: for all î ¤ R, we have { |x| := X -X if x ≥ 0 if x < 0 Prove that for all x ER, we have x² ≥ 0. 4. Let x be a real number. Prove that if x² = x, then x < 2. 5. Prove that if n is an integer, then n² + 3n+1 is odd. 6. Let a, ,b be integers. Prove that if a + b is even, then a - b is even. 7. Let a, b be integers. Prove that if ab is even, then a or b is even.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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For this homework, you can use the following facts without citing. 1. Basic algebra, such as 2+2 = 4, 1-3-2, 2.4 = 8,0 x = 0, 1.x = x, 2.x = x+x. 2. For all integers x, y, z, we have the Associative Law: (x + y) + z = x + (y + z) and (xy)z = x(yz); thus we are allowed to write x + y + z and xyz. We also have the Commutative Law: x +y = y + x and xy = yx. 3. Distributive law: For all a, b, c € C, we have (a + b)(c + d) = ac + ad + bc + bd. In particular, (a + b)² = a² + 2ab +6². 4. Let a, b, c be real numbers with b> c. Then a + b > a+c. 5. Let a, b, c be real numbers with a > 0 and b> c. Then ab > ac. Now here are some definitions / facts that you need to cite when used. The citing instructions are given after the list. 1. An integer n is even if there exists an integer k such that n = 2k. 2. An integer n is odd if there exists an integer k such that n = 2k + 1. 3. All integers are even or odd. 4. If a, b € C and a, b ‡ 0, then ab ‡ 0. In particular, if a² = 0, then a = 0. 5. A real number x is positive x>0, and negative if x < 0. 6. All real numbers are either positive, negative, or 0. When you use any of these, cite "HW3 Facts" and the numbering, or the name of what you are citing (the bold words). You can also use the statement of the previous questions to solve later ones, by citing "HW3 Q (problem number)". However, you are not allowed to use other facts you know. Here is an example. Proposition 1. If n is an integer, then 2n is an even integer. Proof. Let n be an integer. Then there exists an integer k, namely k = n, such that 2n = 2k. By HW2 Fact 6, we have shown that 2n is an even integer. 1. Let n be an integer. Prove that the additive inverse (see Fact 3) of n is unique. (Side note: As a result, it makes sense to say "the" additive inverse. We should NOT use the word "the" unless it is unique. )
Transcribed Image Text:For this homework, you can use the following facts without citing. 1. Basic algebra, such as 2+2 = 4, 1-3-2, 2.4 = 8,0 x = 0, 1.x = x, 2.x = x+x. 2. For all integers x, y, z, we have the Associative Law: (x + y) + z = x + (y + z) and (xy)z = x(yz); thus we are allowed to write x + y + z and xyz. We also have the Commutative Law: x +y = y + x and xy = yx. 3. Distributive law: For all a, b, c € C, we have (a + b)(c + d) = ac + ad + bc + bd. In particular, (a + b)² = a² + 2ab +6². 4. Let a, b, c be real numbers with b> c. Then a + b > a+c. 5. Let a, b, c be real numbers with a > 0 and b> c. Then ab > ac. Now here are some definitions / facts that you need to cite when used. The citing instructions are given after the list. 1. An integer n is even if there exists an integer k such that n = 2k. 2. An integer n is odd if there exists an integer k such that n = 2k + 1. 3. All integers are even or odd. 4. If a, b € C and a, b ‡ 0, then ab ‡ 0. In particular, if a² = 0, then a = 0. 5. A real number x is positive x>0, and negative if x < 0. 6. All real numbers are either positive, negative, or 0. When you use any of these, cite "HW3 Facts" and the numbering, or the name of what you are citing (the bold words). You can also use the statement of the previous questions to solve later ones, by citing "HW3 Q (problem number)". However, you are not allowed to use other facts you know. Here is an example. Proposition 1. If n is an integer, then 2n is an even integer. Proof. Let n be an integer. Then there exists an integer k, namely k = n, such that 2n = 2k. By HW2 Fact 6, we have shown that 2n is an even integer. 1. Let n be an integer. Prove that the additive inverse (see Fact 3) of n is unique. (Side note: As a result, it makes sense to say "the" additive inverse. We should NOT use the word "the" unless it is unique. )
2. Recall that determinant is a function from the set of 2 x 2 matrices over R to the set of real numbers: : {(ad) | a,b,c,dER det : det 7. Let (ad) с (a) Prove that for all real number x, there is a 2 x 2 matrix over R such that its determinant is x. (b) Is the matrix in part (a) unique? Prove or provide a counterexample. 3. We define the absolute value function by: for all x € R, we have |x| := X -X Prove that for all x € R, we have x² ≥ 0. 4. Let x be a real number. Prove that if x²: = ad-bc. → R if x ≥ 0 if x < 0 = x, then x < 2. 5. Prove that if n is an integer, then n² + 3n+ 1 is odd. 6. Let a, b be integers. Prove that if a + b is even, then a - b is even. b be integers. Prove that if ab is even, then a or b is even. A₂
Transcribed Image Text:2. Recall that determinant is a function from the set of 2 x 2 matrices over R to the set of real numbers: : {(ad) | a,b,c,dER det : det 7. Let (ad) с (a) Prove that for all real number x, there is a 2 x 2 matrix over R such that its determinant is x. (b) Is the matrix in part (a) unique? Prove or provide a counterexample. 3. We define the absolute value function by: for all x € R, we have |x| := X -X Prove that for all x € R, we have x² ≥ 0. 4. Let x be a real number. Prove that if x²: = ad-bc. → R if x ≥ 0 if x < 0 = x, then x < 2. 5. Prove that if n is an integer, then n² + 3n+ 1 is odd. 6. Let a, b be integers. Prove that if a + b is even, then a - b is even. b be integers. Prove that if ab is even, then a or b is even. A₂
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