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Chapter 1: Problem 21

Use predicates, quantifiers, logical connectives, and mathematical operators to express the statement that every positive integer is the sum of the squares of four integers.

Short Answer

Expert verified
鈭�n 鈭� 鈩� 鈭僡, b, c, d 鈭� 鈩� (n = a^2 + b^2 + c^2 + d^2).

Step by step solution

01

Identify the main components of the statement

The statement claims that every positive integer can be expressed as the sum of the squares of four integers. Let's denote positive integers by the variable n.
02

Introduce predicates and variables

Define predicates and variables: Let n be a positive integer. Define a predicate P(n) to express that the positive integer n is the sum of the squares of four integers. We need to find integers a, b, c, and d such that n = a^2 + b^2 + c^2 + d^2.
03

Write the mathematical expression using quantifiers

To express 'every positive integer,' we use the universal quantifier 鈭�. To express 'there exist integers a, b, c, and d,' we use the existential quantifier 鈭�.
04

Construct the logical expression

Combine the predicates, quantifiers, and mathematical operators: 鈭�n 鈭� 鈩� 鈭僡, b, c, d 鈭� 鈩� (n = a^2 + b^2 + c^2 + d^2).
05

Verify the constructed expression

Check if the logical expression correctly conveys the original statement: Every positive integer n can be expressed as the sum of the squares of four integers a, b, c, and d.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantifiers
Quantifiers are symbols or words used in mathematics to express the quantity of objects that satisfy certain conditions.
The two most common types of quantifiers are the universal quantifier and the existential quantifier.
  • The universal quantifier ( 鈭� ) means 'for all' or 'every.' For example, 鈭�x 鈭� 鈩� means 'for every x in the set of natural numbers.'
  • The existential quantifier ( 鈭� ) means 'there exists.' For instance, 鈭儁 鈭� 鈩� means 'there exists a y in the set of integers.'
In our exercise, we use 鈭�n 鈭� 鈩� to state that for every positive integer n, there exist four integers (a, b, c, d) such that n equals the sum of their squares.
Logical Connectives
Logical connectives help in forming complex logical expressions from simpler ones.
They include operators such as AND ( 鈭� ), OR ( 鈭� ), NOT ( 卢 ), and IMPLIES ( 鈫� ).
  • AND ( 鈭� ) connects two statements that both must be true. For example, P 鈭� Q means both P and Q are true.
  • OR ( 鈭� ) means at least one of the statements must be true. For example, P 鈭� Q means either P, Q, or both are true.
  • NOT ( 卢 ) negates a statement. For example, 卢P means P is not true.
In mathematical expressions, these connectives allow us to build more complex and precise statements. Though our specific exercise does not heavily use connective operators, understanding them helps in logical structuring of mathematical statements.
Mathematical Operators
Mathematical operators are symbols that represent operations applied to numbers.
These include addition (+), subtraction (-), multiplication (脳), division (梅), and exponentiation (^).
For example, in the equation 3 + 5 = 8, the plus (+) symbol is the operator.
In the sum of squares task, we use the exponentiation operator ( ^ ), where a^2 represents 'a squared.' Also, we use the addition operator (+) to sum these squares.
For instance, in n = a^2 + b^2 + c^2 + d^2, the operators help in forming the sum of four squared integers.
Sum of Squares
The sum of squares refers to the sum obtained when squaring individual numbers and adding them together.
For any given numbers a, b, c, and d, their squares are a^2, b^2, c^2, and d^2.
The sum of squares is then written as a^2 + b^2 + c^2 + d^2.
This concept is useful in various branches of mathematics.
  • It appears in the Pythagorean theorem when dealing with right triangles.
  • It's a common concept in statistics for calculating variance.
In our exercise, we express every positive integer n as a sum of squares of four integers:
n = a^2 + b^2 + c^2 + d^2. Thus, understanding the sum of squares is essential for grasping the full solution.

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Most popular questions from this chapter

Prove that there are infinitely many solutions in positive integers \(x, y,\) and \(z\) to the equation \(x^{2}+y^{2}=\) \(z^{2} .\left[\text { Hint: Let } x=m^{2}-n^{2}, y=2 m n, \text { and } z=m^{2}+n^{2}\right.\) where \(m\) and \(n\) are integers. \(]\)

Express the negations of these propositions using quantifiers, and in English. a) Every student in this class likes mathematics. b) There is a student in this class who has never seen a computer. c) There is a student in this class who has taken every mathematics course offered at this school. d) There is a student in this class who has been in at least one room of every building on campus.

Use a proof by exhaustion to show that a tiling using dominoes of a \(4 \times 4\) checkerboard with opposite corners removed does not exist. [Hint: First show that you can assume that the squares in the upper left and lower right corners are removed. Number the squares of the original checkerboard from 1 to \(16,\) starting in the first row, moving right in this row, then starting in the leftmost square in the second row and moving right, and so on. Remove squares 1 and \(16 .\) To begin the proof, note that square 2 is covered either by a domino laid horizontally, which covers squares 2 and \(3,\) or vertically, which covers squares 2 and \(6 .\) Consider each of these cases separately, and work through all the subcases that arise. \(]\)

Express each of these statements using logical operators, predicates, and quantifiers. a) Some propositions are tautologies. b) The negation of a contradiction is a tautology. c) The disjunction of two contingencies can be a tautology. d) The conjunction of two tautologies is a tautology.

Let \(Q(x)\) be the statement " \(x+1>2 x\) . If the domain consists of all integers, what are these truth values? $$ \begin{array}{llll}{\text { a) }} & {Q(0)} & {\text { b) } Q(-1)} & {\text { c) }} \quad {Q(1)} \\ {\text { d) }} & {\exists x Q(x)} & {\text { e) } \quad \forall x Q(x)} & {\text { f) } \quad \exists x \neg Q(x)}\end{array} $$ g) \(\quad \forall x \neg Q(x)\)
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