# Propositional Equivalences

This category shows the basis of propositional equivalences and solution to text-book exercises to use as a guide to solve other exercises.

## 8. Use De Morgan’s laws to find the negation of each of the following statements.

De Morgan’s Laws: ¬(p∨q)≡¬p∧¬q ¬(p∧q)≡¬p∨¬q As in a previous exercise, if we use propositional variables, we will not make mistakes while solving this type of exercise. a) Kwame will take a job in industry or go to graduate school Let p=”Kwame will take a job in industry” and q=”Kwame will go to graduate school”. ¬(p∨q)≡¬p∧¬q, …

## 11. Show that each conditional statement in Exercise 9 is a tautology without using truth tables

See below the truth table for a conditional statement, and conjunction and disjunctions of two propositions. We will use them to solve this exercise. a) (p∧q)→p From the truth table for the conjunction, we know that if p^q is true, then p and q must be true. If p is true, then (p^q)->p is also …

## 7. Use De Morgan’s laws to find the negation of each of the following statements

De Morgan’s Laws: ¬(p∨q)≡¬p∧¬q ¬(p∧q)≡¬p∨¬q a) Jan is rich and happy The way to solve this type of exercise and make sure not to make a mistake is straightforward: Let p=”Jan is reach” and q=”Jan is happy” The negation of p^q is as follows: ¬(p∧q)≡¬p∨¬q, by the De Morgan Law. Now, we can write it …

## Show that p↔ q and (p∧q)∨(¬p∧¬q) are logically equivalent

To prove that two compound propositions are logically equivalent, we need to prove that they have the same truth values. We can do this in two ways: (1) by creating the truth table for both propositions and comparing the truth values or (2) by using a series of proven logical equivalences that allows concluding the …

## Use a truth table to verify the first De Morgan law

To prove the first De Morgan law, we need to use the truth table for conjunctions, disjunctions, and negation of propositions. Truth tables for conjunction, disjunctions and negation Now we can start solving the exercise. First De Morgan Law The following propositional equivalence is the First De Morgan Law. ¬(p ∧ q) ≡ ¬p ∨ …

## Use truth tables to verify the associative laws

To prove the associative laws, we need to use the truth table for conjunctions and disjunctions. Truth tables for conjunction and disjunctions Now we can start solving the exercise. a) (p∨q)∨r ≡p∨(q∨r) As in previous examples, let’s examine what will be the structure of the truth table we must create. There are three propositional variables, …

## Show that ¬(¬p) and p are logically equivalent

As usual, to solve this type of exercise we should look at the logical operators. In this case, we only have negation (¬), so let’s start by having the truth table for that specific operator. Truth table for the negation (¬) of a proposition Solution to the exercise If we analyze what we have to …

## Use truth tables to verify these equivalences

In this example, we should use a truth table to verify each propositional equivalence. Because in this case the exercises only involve the use of conjunctions (∧) and disjunctions (∧), we start by having at hand (or in memory) the truth tables for those logical operators. Truth tables for conjunction and disjunction a) p∧T≡p In …

## Show that p → q and ¬q → ¬p are logically equivalent

The solution to this exercise is straightforward by constructing the truth table. By looking at the truth table for the two compound propositions p → q and ¬q → ¬p, we can conclude that they are logically equivalent because they have the same truth values (check the columns corresponding to the two compound propositions) Related …

## What are propositional equivalences in Discrete Mathematics?

Propositional equivalences are used extensively in the construction of mathematical arguments. By using these equivalences, we can substitute propositions with other propositions with the same truth value. This proves to be very useful in different types of situations. Two compound propositions p and q are logically equivalent if p ↔ q is a tautology. The …