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Computer Engineering Concepts- Home
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- 3 - Principles of Logic
- 3.1 Propositions
3.1 Propositions
Logic is the analysis of statements that are either true or false; such statements are called propositions. These propositional statements can be in the form of sentences, or in the form of mathematical statements.
Definition: A proposition is a statement that is either true or false, but not both.
Examples of such statements can be found in our everyday language, and we generally use such statements or sentences without giving it much thought.
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Example I: Determine if the statement “Today is Monday” is a proposition. Solution: This statement is either true or false depending on the day of the week you are reading this statement. If the day is Monday then the statement would be true (T), and if today is not Monday then the statement would be false (F). The option of it being both true and false is not possible. Therefore the statement qualifies as a proposition. |
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Example II: Determine if the statement “Read the book” is a proposition. Solution: In this example the statement is not a proposition because there is no concept of true or false attached to the statement. |
From the above two examples it can be seen that not all statements are propositions. The state of a proposition can be either True (T) or False (F), and since there are only two possible states this can be represented in terms of 1 and 0. By convention True (T) is assigned a value of 1, and False is assigned a value of 0 (T=1 and F=0). This is called the truth value of the proposition. Logic, like binary numbers, is of a Boolean nature due to the fact that it has only two states, true and false.
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Example III: Determine if the expression x + 3 > 5 is a proposition, when x =6. Solution: This mathematical statement is another example of a proposition, and in this case it has a truth value of 1 because it is a true statement. It should be noted that if the value of x is not specified the statement would not be a proposition because the truth value cannot be determined without a given value for the variable x. |
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Example IV: Determine the truth value of the following propositions.
Solution:
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3.1 Practice Questions 1. Which of the following statements is a proposition? a. The weather is hot b. Tommy likes cars c. x + 4 - 23 d. Go to school 2. Determine the truth value of the following propositions? a. The temperature is below zero. b. Canada is a tropical country. c. x + 7 > 12, if x = 7 d. 45/3 > 17
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