What rules of inference are used in this famous argument? “All men are mortal. Socrates is a man. Therefore, Socrates is mortal.” 

Rules of inference /

As in the previous examples, first, we need to know the rules of inferences. In this example, you need to realize that you must use predicates instead of propositional variables. Because of that, we also need the rules of inference for quantified statements. Let’s use predicates to represent the argument, then we will apply the […]

What rules of inference are used in this famous argument? “All men are mortal. Socrates is a man. Therefore, Socrates is mortal.”  Read More »

Use rules of inference to show that the hypotheses “Randy works hard,” “If Randy works hard, then he is a dull boy,” and “If Randy is a dull boy, then he will not get the job” imply the conclusion “Randy will not get the job.” 

Rules of inference /

To solve exercises where you have to apply rules of inference, the first step is to know the rules of inference. Once we know the rules, we can then start to apply them. So, let’s start solving the exercise. “Randy works hard,” “If Randy works hard, then he is a dull boy,” and “If Randy

Use rules of inference to show that the hypotheses “Randy works hard,” “If Randy works hard, then he is a dull boy,” and “If Randy is a dull boy, then he will not get the job” imply the conclusion “Randy will not get the job.”  Read More »

Let P(x) be the statement “x = x2.” If the domain consists of the integers, what are these truth values?

Predicates and quantifiers /

To answer this question, we should substitute the values of x in the predicate. Once we do that, the predicate is transformed into a proposition, and we can state the truth value. So, let’s start answering the questions. a) P(0) P(x): “x=x2” P(0): “0=02=0” Answer: true. b) P(1) P(x): “x=x2” P(1): “1=12=1” Answer: true. c)

Let P(x) be the statement “x = x2.” If the domain consists of the integers, what are these truth values? Read More »

Use truth tables to verify the associative laws

Propositional Equivalences /

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,

Use truth tables to verify the associative laws Read More »

For each of these sentences, state what the sentence means if the logical connective or is an inclusive or (that is, a disjunction) versus an exclusive or. Which of these meanings of or do you think is intended?

Propositional Logic /

As usual, we should start with the appropriate definitions. Remember that in mathematics, we must know the definitions before we can start trying to solve a problem. Definitions Let p and q be propositions. The disjunction of p and q, denoted by p ∨ q, is the proposition “p or q.” The disjunction p ∨

For each of these sentences, state what the sentence means if the logical connective or is an inclusive or (that is, a disjunction) versus an exclusive or. Which of these meanings of or do you think is intended? Read More »

Propositional logic Exercise 11. Write these propositions using p and q and logical connectives (including negations)

Propositional Logic /

Solve the following exercise. 11. Let p and q be the propositions p: It is below freezing. q: It is snowing. Write these propositions using p and q and logical connectives (including negations). a) It is below freezing and snowing.b) It is below freezing but not snowing.c) It is not below freezing and it is

Propositional logic Exercise 11. Write these propositions using p and q and logical connectives (including negations) Read More »