Let’s refresh the relevant definitions for this exercise. Definition: “A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.” Definition: “A relation R on a set A is called reflexive if (a,a) ∈ R for every element a ∈ A.” Definition: “A relation R on a set […]
As usual, let’s start with the definitions. Definition: “The compound propositions p and q are called logically equivalent if p ↔ q is a tautology. The notation p ≡ q denotes that p and q are logically equivalent.” Notice that if p ↔q is a tautology, p and q have the same truth values. Definition:
I found out that some students want to solve this type of exercise, where they need to answer if a relation is an equivalence relation, but they don’t know what an equivalence relation is. By doing this, you have a high probability of answering wrong. So, let’s start with the definitions. Definition: “A relation on
a) {(a,b)|a>b}? To answer this exercise, let’s find out how many 1s will have in each row. Then, we add them up. The last row (row 100) will have ninety-nine 1s because 99 numbers are less than 100 in the set A={1,2,3,…,100}. The second last row (row 99) will have ninety-eight 1s as 98 numbers
This is an interesting type of exercise. To make easier this exercise, let’s write an irreflexive relation, represent it using a matrix, and find out what are the characteristics of such a matrix. But first, let’s refresh what an irreflexive relation is. Definition: “A relation R on the set A is irreflexive if for every
In this case, we are given the matrix that represents the relation, and we need to write the pairs that belong to the relation. It is the opposite of this exercise. a) Matrix 1 The approach to solving this type of exercise is to find where there is a 1 in the matrix and write
Recap: A relation R can be represented as a zero-one matrix. If (ai,bj)∈R, then we write a 1 in the matrix in the position (i,j). Otherwise, we write a 0. The size of the matrix will be determined by the size of the set. If you have a relation R on the set A={1,2,3}, the
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
Show that p↔ q and (p∧q)∨(¬p∧¬q) are logically equivalent Read More »
Definition: “A relation R on a set A is called reflexive if (a,a) ∈ R for every element a ∈ A.” Definition: “A relation R on a set A is called symmetric if (b,a) ∈ R whenever (a,b) ∈ R, for all a,b ∈ A. A relation R on a set A such that for
As usual, let’s start with relevant definitions. Definition: “A relation R on a set A is called reflexive if (a,a) ∈ R for every element a ∈ A.” Definition: “A relation R on a set A is called symmetric if (b,a) ∈ R whenever (a,b) ∈ R, for all a,b ∈ A. A relation R
