To solve this exercise, we have to options. We can answer using a state table or state diagram. First, we need to consider that to be in the final state, we need first to start from S0, and recognize a 0, and then a 1. This tells us that we should have at least three […]
“Rules of inferences” is an important topic in Discrete Mathematics. We can use a rule (s) of inferences to prove if an argument is valid or not. We can also use rules of inference to produce valid arguments. In mathematics, an argument is a sequence of statements. We have a valid argument if and only
What rule of inference is used in each of these arguments? Read More »
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
Show that p → q and ¬q → ¬p are logically equivalent Read More »
A classic exercise in Discrete Mathematics is to negate a given proposition. Here, I’ll explain two things to consider after seeing how many students have problems solving this type of exercise. Background A proposition is a sentence that states a fact. It is True or False but not both. An example of a proposition is
Negating propositions Read More »
Introduction The universal quantifier is denoted by ∀”. An example of how to interpret it is as follows: The statement “∀x P(x) is true if P(x) holds for all x in the universe of discourse. On the other hand, if you can find x such that P(x) is false, then the statement “∀x P(x) is
Predicates and quantifiers: Exercise 9 Read More »
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
What are propositional equivalences in Discrete Mathematics? Read More »
Truth tables are very important in Discrete Mathematics. You can use them to calculate the truth value of (compound) propositions, to determine if a compound proposition is a tautology or a contradiction, and also to verify whether two propositions are logically equivalent. Let’s see an example of how to create a truth table for the
Create the truth table for the compound proposition (p∨¬q)->(p∧q) Read More »
The purpose of this post is to explain how to solve this type of exercise. Suppose that Smartphone A has 256 MB RAM and 32 GB ROM, and the resolution of its camera is 8 MP; Smartphone B has 288 MB RAM and 64 GB ROM, and the resolution of its camera is 4 MP,
Propositional Logic: Exercise 6 Read More »
Mathematical induction is a powerful tool we should have in our toolbox. Here I’ll explain the basis of this proof method and will show you some examples. The theory behind mathematical induction You can be surprised at how small and simple the theory behind this method is yet so powerful. In general, mathematical induction can
Mathematical induction with examples Read More »
Sets is one of the basic structures in Discrete Mathematics. Here, you will find the solution to 5 set exercises from one of the more used textbooks in this topic. Exercise 1. List the members of the following sets In this case, we have to use the roaster method. ” There are several ways to
Basic structures: sets. Solutions to set exercises from the textbook Read More »
