Discrete Mathematics

Use a proof by contradiction to prove that the sum of an irrational number and a rational number is irrational

As in previous cases, we start by expressing the conjecture as a conditional statement. If m is irrational and n is rational, then m+n is irrational. In this case, we must use proof by contradiction. Remember that a contradiction is a compound proposition that is always false. To give a proof by contradiction of a …

Use a proof by contradiction to prove that the sum of an irrational number and a rational number is irrational Read More »

For each of these collections of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises. 

As with the previous examples, there are two main things we need to solve this exercise. First, we need to know the rules of inference so we can apply them. Second, we need to identify whether we must use propositional variables or predicates. The first step will always be to translate the argument using propositional …

For each of these collections of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises.  Read More »

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

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.” 

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?

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

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 »