First, let’s refresh the concept of direct proof. “A direct proof of a conditional statement p → q is constructed when the first step is the assumption that p is true; subsequent steps are constructed using rules of inference, with the final step showing that q must also be true. A direct proof shows that […]
3. Show that the square of an even number is an even number using a direct proof Read More »
The following will serve as a guide to answer this exercise: a) ∀x(C(x)→F(x)) All comedians are funny. b) ∀x(C(x)∧F(x)) Every person is funny and a comedian. A shortened way is every person is a funny comedian. c) ∃x(C(x)→F(x)) There exists at least one person that, if that person is a comedian, then the person is
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,
8. Use De Morgan’s laws to find the negation of each of the following statements. Read More »
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
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
7. Use De Morgan’s laws to find the negation of each of the following statements Read More »
Relevant definitions: “A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a ∈ A to denote that a is an element of the set A. The notation a∉A denotes that a is not an element of the set A.”
8. For each of the sets in Exercise 7, determine whether {2} is an element of that set Read More »
Relevant definitions: THE EMPTY SET: “There is a special set that has no elements. This set is called the empty set, or null set, and is denoted by ∅. The empty set can also be denoted by { } (that is, we represent the empty set with a pair of braces that encloses all the
10. Determine whether these statements are true or false Read More »
Relevant definition: “The set A is a subset of B if and only if every element of A is also an element of B. We use the notation A ⊆ B to indicate that A is a subset of the set B.” Discrete Mathematics and its Applications by Rosen. To make sure that we check
Relevant definitions: THE EMPTY SET: “There is a special set that has no elements. This set is called the empty set, or null set, and is denoted by ∅. The empty set can also be denoted by { } (that is, we represent the empty set with a pair of braces that encloses all the
9. Determine whether each of these statements is true or false Read More »
Relevant definitions: “A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a ∈ A to denote that a is an element of the set A. The notation a∉A denotes that a is not an element of the set A.”
7. For each of the following sets, determine whether 2 is an element of that set Read More »
