Math Problem Statement

Translate English statements about integers and days into logical expressions.

Solution

Let's break down the problem you shared:

You are working with two statements:

  • x: "an integer is even."
  • y: "it’s Tuesday."

We need to translate the statements a) through e) into logical expressions based on x and y, and then select the correct logical symbol from the choices provided.

Here are the translations of each statement into logical expressions:

a) An integer is odd and it’s not Tuesday.

  • "Odd" means not even, so ¬x.
  • "It’s not Tuesday" means ¬y.
  • So, the translation is: ¬x ∧ ¬y

b) An integer is odd or it’s not Tuesday.

  • "Odd" is ¬x, and "it’s not Tuesday" is ¬y.
  • "Or" is a disjunction.
  • So, the translation is: ¬x ∨ ¬y

c) An integer is even and it’s Tuesday.

  • "Even" is x and "it’s Tuesday" is y.
  • "And" is a conjunction.
  • So, the translation is: x ∧ y

d) An integer is even if and only if it’s Tuesday.

  • This is a biconditional statement, meaning both must be true or both false.
  • So, the translation is: x ↔ y

e) An integer is odd or it’s Tuesday.

  • "Odd" is ¬x, and "it’s Tuesday" is y.
  • "Or" is a disjunction.
  • So, the translation is: ¬x ∨ y

Now, match each logical expression to the provided symbols:

  • a) ¬x ∧ ¬y: Corresponds to the 3rd symbol from the left.
  • b) ¬x ∨ ¬y: Corresponds to the 6th symbol from the left.
  • c) x ∧ y: Corresponds to the 1st symbol from the left.
  • d) x ↔ y: Corresponds to the 7th symbol from the left.
  • e) ¬x ∨ y: Corresponds to the 5th symbol from the left.

Let me know if you need more details, or if you have any questions.

Related questions:

  1. What is the difference between a conjunction and a disjunction in logic?
  2. How do you negate a biconditional statement?
  3. Can a statement be both a conjunction and a disjunction? Explain.
  4. How would the logic change if the integer were required to be even?
  5. How do truth tables help in evaluating logical expressions?

Tip: When translating word problems into logical expressions, pay close attention to the keywords "and," "or," "not," and "if and only if." These directly correspond to logical operations like conjunction, disjunction, negation, and biconditionals.

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Math Problem Analysis

Mathematical Concepts

Propositional Logic
Logical Operators
Biconditional Statements

Formulas

¬x
¬y
∧ (conjunction)
∨ (disjunction)
↔ (biconditional)

Theorems

Basic logical equivalences
Biconditional equivalence

Suitable Grade Level

Grades 10-12