This lesson begins our third main topic area in the course, and many consider it to be the heart of both discrete math and computer science -- the study of logic. Logic is the language that allows us to write computer programs that make decisions; it also allows us a framework to think more clearly about the world around us. In our introductory lesson in logic, we'll look at the building blocks of logic, namely propositions and operators that apply to and combine propositions.
Basic objectives: Each student is responsible for gaining proficiency with each of these tasks prior to engaging in class discussions, through the use of the learning resources (below) and through the working of exercises (also below).
- Write the truth tables for conjunction, disjunction, negation, the conditional operator, and the biconditional operator.
- Write the converse of a conditional statement.
- Express logical propositions that are written in English using symbolic form, and translate propositions from symbolic form to English.
Advanced objectives: The following objectives are the subject of class discussion and further work; they should be mastered by each student during and following class discussions.
- Write the contrapositive of a conditional statement.
- Determine the condition(s) under which a conditional statement is false.
To gain proficiency in the learning objectives, use the following resources. You may include other resources if you wish, in addition to or in replacement of the following.
Textbook: In Applied Discrete Structures, read Section 3.1. Make sure to read actively, working through examples and activities as you go. Also: Look up the definition of the contrapositive of a conditional statement. Here is one source.
Video: Watch the following videos. Some things to note: (1) The first video uses the term "statement" whereas we will use the term "proposition". (2) The second video uses T and F for true and false whereas your book uses the bits 1 and 0 for true and false respectively. (This bit notation is common in CS and engineering.)
- Statements and Non-Statements (5:37)
- Introduction to Logic, watch only up through 4:33 and you don't need to know about quantifiers of statements yet.
The following exercises are to be done during and following your reading and viewing of the resources. Work these out on paper and then enter the responses into the appropriate submission form (see Submission Instructions) by the deadline. You will receive a mark of Pass if each item response shows a good-faith effort to be right and is submitted prior to the deadline.
Let the variable
- Express the English statement "I like discrete structures and I will pass this course" in symbolic form, using the variable names above and some of the symbols for logical operators.
- Consider the proposition
$d \rightarrow (c \wedge s)$ . Rewrite this proposition as a correct and legible English sentence. - Consider the proposition "If I complete this assignment, then I will pass this class." Write both the converse and the contrapositives of this proposition.
- Consider "If I complete this assignment, then I will pass this class." Under what condition will this statement be false?
- Can you think of a conditional (if-then) statement which is true, but whose converse is false? If so, write that statement in the blank. If not, write "none found".
Submit your responses using the form at this link: http://bit.ly/1LLHFxX