tmua logic and proof
Flashcards: Logic and Proof for TMUA
Section 1: Statements and Truth Values
Flashcard 1 Q: What is a mathematical statement?
A: A sentence that is either definitely true or definitely false, but not both. This is based on the law of the excluded middle.
Flashcard 2 Q: What is the truth value of a statement?
A: Whether the statement is true or false. For example, the truth value of "2 is an even number" is true, while the truth value of "2 is an odd number" is false.
Flashcard 3 Q: What are logically equivalent statements?
A: Statements that have the same truth values in all circumstances. For example, "Today is Tuesday" and "Today is the day after Monday" are logically equivalent.
Flashcard 4 Q: What does the document mean by "truth tables"?
A: Truth tables list all possible truth values of statements and their combinations. For instance, a truth table for "A and B" evaluates the combined truth based on the truth of A and B.
Flashcard 5 Q: How are truth values handled in conditional expressions?
A: If the condition cannot be evaluated as true or false, it is not considered a valid mathematical statement.
Section 2: Combining Statements
Flashcard 6 Q: What is the truth table for "not A"?
A:
Anot A | |
T | F |
F | T |
Flashcard 7 Q: Define "A and B". When is it true?
A: "A and B" is true only when both A and B are true.
Truth table:
A | B | A and B |
T | T | T |
T | F | F |
F | T | F |
F | F | F |
Flashcard 8 Q: Define "A or B" in mathematical logic.
A: "A or B" means "A is true, or B is true, or both are true" (inclusive or).
Truth table:
A | B | A or B |
T | T | T |
T | F | T |
F | T | T |
F | F | F |
Flashcard 9 Q: How do you negate "A and B"?
A: The negation of "A and B" is "not A or not B." Truth table:
A | B | A and B | not (A and B) |
T | T | T | F |
T | F | F | T |
F | T | F | T |
F | F | F | T |
Flashcard 10 Q: What is the logical equivalence of "not (A or B)"?
A: "Not (A or B)" is equivalent to "not A and not B."
Section 3: If Statements
Flashcard 11 Q: What does "if A then B" mean in logic?
A: If A is true, then B must also be true. If A is false, B can be either true or false. Truth table:
A | B | if A then B |
T | T | T |
T | F | F |
F | T | T |
F | F | T |
Flashcard 12 Q: What is the equivalent logical expression for "if A then B"?
A: "if A then B" is logically equivalent to "not A or B."
Flashcard 13 Q: How are "if A then B" and "B if A" related?
A: These two statements are logically equivalent and mean the same thing.
Flashcard 14 Q: What is the difference between "if A then B" and "A only if B"?
A: "A only if B" means "if not B, then not A," which is the contrapositive of "if A then B."
Section 4: Necessary and Sufficient Conditions
Flashcard 15 Q: What is a necessary condition?
A: A condition that must be true for a statement to hold. For example, "Being divisible by 4" is a necessary condition for "Being divisible by 8."
Flashcard 16 Q: What is a sufficient condition?
A: A condition that guarantees the truth of a statement. For example, "Being divisible by 8" is a sufficient condition for "Being divisible by 4."
Flashcard 17 Q: Can a condition be necessary but not sufficient? Give an example.
A: Yes. For example, "Having an angle in common" is necessary but not sufficient for two triangles to be similar.
Flashcard 18 Q: What is a necessary and sufficient condition?
A: A condition that is both required and guarantees the truth of a statement. For example, "A triangle is equilateral if and only if all its sides are equal."
Section 5: Proof Techniques
Flashcard 19 Q: What is a direct proof?
A: A logical argument that shows a statement is true by a sequence of deductive steps. Example: Proving the sum of two even numbers is even.
Flashcard 20 Q: What is proof by contradiction?
A: Assuming the opposite of what you want to prove, deriving a contradiction, and concluding that the original statement must be true.
Flashcard 21 Q: What is proof by cases?
A: Dividing a problem into separate cases and proving the statement for each case.
Flashcard 22 Q: What is disproof by counterexample?
A: Demonstrating that a general statement is false by providing a specific example where it does not hold. Example: "All prime numbers are odd" is disproven by 2.
Section 6: Converse and Contrapositive
Flashcard 23 Q: What is the converse of "if A then B"?
A: "If B then A."
Flashcard 24 Q: What is the contrapositive of "if A then B"?
A: "If not B then not A."
Note: The contrapositive is logically equivalent to the original statement.
Flashcard 25 Q: What is the inverse of "if A then B"?
A: "If not A, then not B."
Note: The inverse is not necessarily logically equivalent to the original statement.
Exercises
Flashcard 26 Q: Negate the statement: "All swans are white."
A: "There exists at least one swan that is not white."
Flashcard 27 Q: What is the contrapositive of "If a number is divisible by 4, then it is even"?
A: "If a number is not even, then it is not divisible by 4."
Flashcard 28 Q: What is the converse of "If a triangle is equilateral, then it is isosceles"?
A: "If a triangle is isosceles, then it is equilateral."
Flashcard 29 Q: Write the contrapositive of "If x > 2, then x^2 > 4."
A: "If x^2 <= 4, then x <= 2."
Flashcard 30 Q: What is the contrapositive of "If a number is prime, then it has exactly two distinct positive divisors"?
A: "If a number does not have exactly two distinct positive divisors, then it is not prime."