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Chapter 1: Problem 19

Express each of these statements using mathematical and logical operators, predicates, and quantifiers, where the domain consists of all integers. a) The sum of two negative integers is negative. b) The difference of two positive integers is not necessarily positive. c) The sum of the squares of two integers is greater than or equal to the square of their sum. d) The absolute value of the product of two integers is the product of their absolute values.

Short Answer

Expert verified
a) \( \forall x \forall y ( (N(x) \text{ and } N(y)) \rightarrow N(x + y) )\) b) \( \forall x \forall y ((P(x) \text{ and } P(y)) \rightarrow eg(P(x - y)) )\) c) \( \forall x \forall y (S(x) + S(y) \rightarrow S(x + y))\) d) \( \forall x \forall y (A(x \times y) = A(x) \times A(y))\)

Step by step solution

01

Express Statement (a) - Sum of Two Negative Integers

For statement (a), we need to express that the sum of two negative integers is always negative. Let's denote the predicate for 'is negative' as \(N(x)\), which means \(x < 0\). Then we can write: \( \forall x \forall y ( (N(x) \text{ and } N(y)) \rightarrow N(x + y) )\) This states that for all integers \(x\) and \(y\), if both \(x\) and \(y\) are negative, then their sum \(x + y\) is also negative.
02

Express Statement (b) - Difference of Two Positive Integers

For statement (b), we need to show that the difference of two positive integers is not necessarily positive. Let's denote the predicate for 'is positive' as \(P(x)\), which means \(x > 0\). Then we can write: \( \forall x \forall y ((P(x) \text{ and } P(y)) \rightarrow eg(P(x - y)) )\) This states that for all integers \(x\) and \(y\), if both \(x\) and \(y\) are positive, their difference \(x - y\) is not necessarily positive (not always true).
03

Express Statement (c) - Sum of Squares

For statement (c), we need to express that the sum of the squares of two integers is greater than or equal to the square of their sum. Let the predicate squares be denoted by \(S(x) = x^2\). Then we can write: \( \forall x \forall y (S(x) + S(y) \times \rightarrow S(x + y))\) This means for all integers \(x\) and \(y\), the sum of their squares \(x^2 + y^2\) is always greater than or equal to the square of their sum \((x + y)^2\).
04

Express Statement (d) - Absolute Value of Product

For statement (d), we need to express that the absolute value of the product of two integers is the product of their absolute values. Let the predicate for 'absolute value' be \( A(x) \t = |x|\). Then we write: \( \forall x \forall y (A(x \times y) = A(x) \times A(y))\) This means for all integers \(x\) and \(y\), the absolute value of their product \(|xy|\) is equal to the product of their absolute values \(|x| \times |y|\).

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Key Concepts

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

logical operators
Logical operators are fundamental to mathematical logic. They help in forming statements that combine or modify predicates. The most common logical operators are:
  • AND ( 鈭� ): Both conditions must be true for the overall statement to be true.
  • OR ( 鈭� ): At least one condition must be true for the overall statement to be true.
  • NOT ( 卢 ): Inverts the truth value of a condition.
  • IMPLY ( 鈫� ): If the first condition is true, then the second must also be true.
In the solutions provided, these operators play key roles. For instance, in step 1, we used the AND ( 鈭� ) operator to state that both integers are negative. Then, we used the IMPLY ( 鈫� ) operator to express that this leads to their sum being negative.
predicates
Predicates are statements that may be true or false depending on the values of their variables. They often involve relations or properties concerning the variables. For example, in the provided solutions:
  • P(x) : 'x is positive' (x > 0).
  • N(x) : 'x is negative' (x < 0).
  • S(x) : 'x squared' (x^2).
  • A(x) : 'absolute value of x' (|x|).
Thus, the predicates help to provide a basis for forming logical statements involving integers. For instance, P(x) and P(y) are used in step 2 to mark that both x and y are positive.
quantifiers
Quantifiers are used to state that expressions are true for all elements of a domain or for some elements. The two main types are:
  • Universal Quantifier ( 鈭� ): This indicates that a predicate is true for all elements in a domain.
  • Existential Quantifier ( 鈭� ): This indicates that there exists at least one element in the domain for which the predicate is true.
In the exercises, universal quantifiers ( 鈭� ) are frequently used to specify that statements hold for all integers. For example, in step 3, we state 鈭� x 鈭� y to mean that for all integers x and y, the sum of their squares is always greater than or equal to the square of their sum.
integer properties
Understanding the properties of integers aids in formulating and verifying logical statements. Important properties include:
  • Commutative Property: Addition and multiplication of integers are commutative (e.g., a + b = b + a).
  • Associative Property: Addition and multiplication of integers are associative (e.g., (a + b) + c = a + (b + c)).
  • Distributive Property: Multiplication distributes over addition (e.g., a(b + c) = ab + ac).
  • Properties of Negatives: The sum, product, or difference of negative integers follows specific rules.
  • Absolute Value: The absolute value | x | represents the non-negative value of x.
For instance, in step 1, the sum of two negative integers following integer properties results in a negative integer (commutative and associative properties ensure this). Similarly, in step 4, the product of the absolute values is addressed by acknowledging that the absolute value property converts any integer to its positive form.

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Most popular questions from this chapter

Exercises \(40-44\) deal with the translation between system specification and logical expressions involving quantifiers. Express each of these system specifications using predicates, quantifiers, and logical connectives. a) Every user has access to an electronic mailbox. b) The system mailbox can be accessed by everyone in the group if the file system is locked. c) The firewall is in a diagnostic state only if the proxy server is in a diagnostic state. d) At least one router is functioning normally if the throughput is between 100 kbps and 500 kbps and the proxy server is not in diagnostic mode.

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.

Exercises \(61-64\) are based on questions found in the book Symbolic Logic by Lewis Carroll. Let P(x), Q(x), R(x), and S(x) be the statements 鈥渪 is a baby,鈥� 鈥渪 is logical,鈥� 鈥渪 is able to manage a crocodile,鈥� and 鈥渪 is despised,鈥� respectively. Suppose that the domain consists of all people. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), R(x), and S(x). a) Babies are illogical. b) Nobody is despised who can manage a crocodile. c) Illogical persons are despised. d) Babies cannot manage crocodiles. e) Does (d) follow from (a), (b), and (c)? If not, is there a correct conclusion?

Express each of these statements using quantifiers. Then form the negation of the statement, so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase "It is not the case that.") a) Some old dogs can learn new tricks. b) No rabbit knows calculus. c) Every bird can fly. d) There is no dog that can talk. e) There is no one in this class who knows French and Russian.

Use rules of inference to show that if \(\forall x(P(x) \vee Q(x))\) \(\forall x(\neg Q(x) \vee S(x)), \quad \forall x(R(x) \rightarrow \neg S(x)),\) and \(\exists x \neg P(x)\) are true, then \(\exists x \neg R(x)\) is true.
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