Math Problem Statement

Let p and q be two statements: p: The antivirus is very good this year. q: Computers are getting spoilt. Give the verbal translation for the following: (a) p ∧ ~q (b) ~p ∨ q (c) ~p ∨ (~p ∨ q) (d) p ⇔ q

Solution

The problem you provided involves logical statements, where:

  • pp: The antivirus is very good this year.
  • qq: Computers are getting spoilt.

Let's go through the verbal translations for the logical expressions one by one:

(a) p¬qp \land \neg q:
"The antivirus is very good this year, and computers are not getting spoilt."

(b) ¬pq\neg p \lor q:
"Either the antivirus is not very good this year, or computers are getting spoilt."

(c) ¬p(¬pq)\neg p \lor (\neg p \lor q):
"This seems to be a typographical issue. I think you meant to write something like ¬p(¬q)\neg p \lor (\neg q), which would translate to:
"Either the antivirus is not very good this year, or computers are not getting spoilt."

(d) p    qp \iff q:
"The antivirus is very good this year if and only if computers are getting spoilt."

Feel free to ask for clarification or further details. Here are some related questions:

  1. How do we interpret conjunctions and disjunctions in logic?
  2. What does "if and only if" mean in logical terms?
  3. How can logical equivalences simplify expressions?
  4. What role do negations play in formal logic?
  5. Can these logical expressions be used in real-life situations like programming or mathematics?

Tip: Practice converting between logical symbols and verbal sentences to become more comfortable with formal logic!

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Math Problem Analysis

Mathematical Concepts

Propositional Logic
Logical Connectives
Conjunction
Disjunction
Negation
Biconditional

Formulas

p ∧ ~q
~p ∨ q
~p ∨ (~p ∨ q)
p ⇔ q

Theorems

Logical Equivalences
Properties of Logical Connectives

Suitable Grade Level

Grades 10-12