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Questions and Answers
What is the value of the expression $p \rightarrow q$ when $p$ is false and $q$ is true?
What is the value of the expression $p \rightarrow q$ when $p$ is false and $q$ is true?
Which statement describes the relationship of $q \rightarrow p$ compared to $p \rightarrow q$?
Which statement describes the relationship of $q \rightarrow p$ compared to $p \rightarrow q$?
What term describes the new statement formed by placing 'if and only if' between two statements?
What term describes the new statement formed by placing 'if and only if' between two statements?
If $p$ is true and $q$ is false, what is the truth value of $p \land q$?
If $p$ is true and $q$ is false, what is the truth value of $p \land q$?
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Which of the following is a correct statement regarding logical connectives?
Which of the following is a correct statement regarding logical connectives?
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What is the truth value of the conjunction p ^ q if p is true and q is false?
What is the truth value of the conjunction p ^ q if p is true and q is false?
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Which of the following statements best defines disjunction?
Which of the following statements best defines disjunction?
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What is the symbol for negation?
What is the symbol for negation?
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Under what condition is the exclusive OR (XOR) operation true?
Under what condition is the exclusive OR (XOR) operation true?
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Given p: '5 is an even integer', what is the truth value of p?
Given p: '5 is an even integer', what is the truth value of p?
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If p is true and q is false, what is the truth value of the statement p → q?
If p is true and q is false, what is the truth value of the statement p → q?
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Which connective represents the implication in logical statements?
Which connective represents the implication in logical statements?
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How many truth values are possible for propositions p and q in a logical statement?
How many truth values are possible for propositions p and q in a logical statement?
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What does Discrete Mathematics primarily deal with?
What does Discrete Mathematics primarily deal with?
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Which of the following is an example of a proposition?
Which of the following is an example of a proposition?
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In logic, what does a truth table help determine?
In logic, what does a truth table help determine?
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What is the correct negation of the statement '2 is positive'?
What is the correct negation of the statement '2 is positive'?
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Which of the following is NOT a proposition?
Which of the following is NOT a proposition?
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Which logical connective is used to negate a statement?
Which logical connective is used to negate a statement?
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Which of the following statements is true?
Which of the following statements is true?
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Which of the following best describes a statement variable?
Which of the following best describes a statement variable?
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Study Notes
Discrete Structure Reviewer
- Continuous Mathematics deals with continuous number lines and real numbers. There is an infinite set of numbers between any two numbers.
- Discrete Mathematics deals with distinct, separate values. There is a countable number of points between any two points.
Logic
- Logic is the study of reasoning. It provides rules and techniques to determine if arguments are valid.
- Example: If Peter solved seven problems correctly, he earned an A grade. Peter earned an A grade. Therefore, Peter solved seven problems correctly.
- Logic uses propositions—statements that are either true (T) or false (F), corresponding to 1 and 0 in digital circuits.
Proposition
- Definition: A proposition is a statement that can be judged as either true or false, but not both.
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Examples of Propositions:
- 4 is an integer.
- 5 is an integer.
- Washington, D.C., is the capital of the USA.
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Non-Propositions:
- "How are you?" (a question)
Statements
- Statements express propositions in language.
- Examples of Statements: "The grass is green" (true), "The Earth is flat" (false).
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Statement vs. Proposition Game:
- Elephants are bigger than mice: Proposition (True)
- 520 < 111: Proposition (False)
- y > 5: Not a proposition (depends on the value of y)
Statement Variables
- Symbols like p, q, r,... represent statements.
Sets and Truth Tables
- Set: A set is a collection of objects, and each object is called an element of the set.
- Truth Table: A truth table is a tool for determining the truth value of propositions and the validity of arguments.
Logical Connectives
- Negation: The negation of a statement (p) is written as ~p and is the opposite of the original statement.
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Examples of Connectives:
- Conjunction (AND): Symbol ^
- Disjunction (OR): Symbol v
- Implication: Symbol →
- Double Implication: Symbol ↔
Conjunction (AND) - Examples
- p: 2 is an even integer.
- q: 7 divides 14.
- p ^ q: 2 is an even integer AND 7 divides 14 (This is true)
Disjunction (OR)
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Consider the following statements:
- p: 5 is an even integer.
- q: 3 is greater than 5.
- p v q: 5 is an even integer OR 3 is greater than 5.
Exclusive OR
- The Exclusive Or (XOR) of two propositions is true when exactly one of the propositions is true. The other proposition must be false.
- Example: A circuit can be ON or OFF, but not both.
Implication
- In implications, two statements are connected by "if...then" to form a new statement.
Biimplication
- Given two statements, a new statement is formed by using the phrase "if and only if".
Statement Formulas
- Examples of statement formulas include (A), (AB), (A→B), and (AB).
- ~, ^, v, <->, → are logical connectives.
- p, q, r, ... are statement variables.
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Description
This quiz covers key concepts in Discrete Mathematics and Logic, including the nature of discrete values and propositions. It also explores reasoning techniques and the validity of arguments. Test your understanding of these fundamental mathematical principles.
