PRELIM DISCRETE MATH Flashcards
1
Q
- What is the output of an AND gate when both inputs are 1?
A. 0
B. 1
C. Either 0 or 1
D. Undefined
A
B. 1 (AND gate outputs 1 when both inputs are 1)
2
Q
- In the statement “Vx(P(x)→Q(x))”, what does the symbol V represent?
A. Existential quantifier
B. Universal quantifier
C. Logical implication
D. Logical conjunction
A
B. Universal quantifier (∀ represents “for all”)
3
Q
- Which logic gate performs the opposite function of an OR gate?
A. AND gate
B. XOR gate
C. NOR gate
D. NAND gate
A
C. NOR gate (NOR is the opposite of OR)
4
Q
- In a direct proof, what is the first step?
A. Assume the conclusion is false
B. Assume the hypothesis is true
C. State the contradiction
D. Write the negation
A
B. Assume the hypothesis is true (First step in direct proof)
5
Q
- What is the output of an XOR gate when both inputs are the same?
A. Always 1
B. Always O
C. Depends on the input value
D. Undefined
A
B. Always 0 (XOR outputs 0 when inputs are the same)
6
Q
- For the statement “3x P(x)”, how would you disprove it?
A. Show P(x) is true for one x
B. Show P(x) is true for all x
C. Show P(x) is false for all x
D. Show P(x) is false for one x
A
D. Show P(x) is false for one x (Disproving an existential quantifier)
7
Q
- In an indirect proof (proof by contradiction), what do we initially assume?
A. The hypothesis is true
B. The conclusion is true
C. The negation of the conclusion.
D. Both hypothesis and conclusion are false
A
The negation of the conclusion (Indirect proof assumes ¬Q)
8
Q
- Which gate is known as a universal gate?
A. OR gate
B. AND gate
C. NAND gate
D. XOR gate
A
C. NAND gate (NAND is a universal gate
9
Q
- What does the existential quantifier (3) assert?
A. The statement is true for all values
B. The statement is true for at least one value
C. The statement is false for all values
D. The statement is false for at least one value
A
B. The statement is true for at least one value (∃ means “exists”)
10
Q
- What is the output of a NOR gate when both inputs are 0?
A. O
B. 1
C. Either 0 or 1
D. Undefined
A
B. 1 (NOR gate outputs 1 when both inputs are 0)
11
Q
- In mathematical logic, what is the contrapositive of “P→Q”?
A. Q→P
B. ~P~Q
C. ~Q→~P
D. P→~Q
A
C. ~Q → ~P (Contrapositive of P → Q)
12
Q
- How many inputs are required for a basic XOR gate?
A. 1
B. 2
C. 3
D. 4
A
B. 2 (XOR gate requires two inputs)
13
Q
- What is the primary purpose of using quantifiers in mathematical logic?
A. To make statements more complex
B. To specify the scope of variables
C. To create truth tables
D. To prove contradictions.
A
B. To specify the scope of variables (Purpose of quantifiers)
14
Q
- Which method of proof assumes a statement is false and arrives at a
contradiction?
A. Direct proof
B. Proof by cases
C. Proof by contradiction
D. Proof by induction
A
C. Proof by contradiction (Assumes false, derives contradiction)
15
Q
- What is the output of a NAND gate when both inputs are 1?
A. 1
В. 0
C. Either 0 or 1
D. Undefined
A
B. 0 (NAND gate outputs 0 when both inputs are 1)
16
Q
- What is a logical statement?
A. A phrase that contains only numbers
B. A declarative sentence that is either true or false
C. A question that requires a yes or no answer
D. An exclamatory sentence expressing emotion
A
B. A declarative sentence that is either true or false (Definition of a logical statement)
17
Q
- Which of the following is a logical connective?
A. Therefore
B. Meanwhile
C. However
D. And
A
D. And (Logical connective)
18
Q
- In a conditional statement “if p then q,” what is the converse?
A. If not p then not q
B. If q then p
C. If not q then not p
D. If p then not q
A
B. If q then p (Converse of “if p then q”)
19
Q
- What is the contrapositive of “If it rains, then the ground is wet”?
A. If it doesn’t rain, then the ground isn’t wet
B. If the ground is wet, then it rains
C. If the ground isn’t wet, then it doesn’t rain
D. If it rains, then the ground isn’t wet
A
C. If the ground isn’t wet, then it doesn’t rain (Contrapositive)
20
Q
- Which rule of inference states that if p→q and q→r are true, then p→r is true?
A. Modus Ponens
B. Hypothetical Syllogism
C. Modus Tollens
D. Disjunctive Syllogism
A
B. Hypothetical Syllogism (If p → q and q → r, then p → r)
21
Q
- What is the inverse of “If you study hard, then you will pass”?
A. If you pass, then you studied hard
B. If you don’t pass, then you didn’t study hard
C. If you don’t study hard, then you won’t pass
D. If you study hard, then you won’t pass
A
C. If you don’t study hard, then you won’t pass (Inverse)
22
Q
D. If you study hard, then you won’t pass 22. Which of the following is a tautology?
A. p^q
B. p V ~p
C. p q
D.p↔q
A
B. p V ~p (Tautology)
23
Q
- In logical equivalence, what does the symbol = represent? A. Exactly equal to
B. Approximately equal to
C. Logically equivalent to
D. Not equal to
A
C. Logically equivalent to (≡ symbol meaning)
24
Q
- What is Modus Ponens?
A. If p→q and ~q, then ~p
B. If p→q and p, then q
C. If p q and q→r, then p→r
D. If pvq and ~p, then q
A
B. If p → q and p, then q (Modus Ponens)
25
25. Which logical connective is represented by the symbol ^?
A. Or
B. If and only if
C. And
D. Not
C. And (∧ represents AND)
26
26. What is the definition of a biconditional statement?
A. A statement that is always false
B. A statement that is always true
C. A statement of the form "p if and only if q"
D. A statement that cannot be determined
C. A statement of the form "p if and only if q" (Biconditional)
27
27. In symbolic logic, what does the symbol represent?
A. And
B. Or
C. Not
D. If then
C. Not (¬ represents negation)
28
28. What is the definition of logical equivalence?
A. Two statements with different truth values
B. Two statements that are both true
C. Two statements with the same truth values under all possible conditions
D. Two statements that are both false
C. Two statements with the same truth values under all possible conditions (Logical equivalence)
29
29. Which rule of inference states that if p→q and ~q are true, then ~p is true?
A. Modus Ponens
B. Addition
C. Conjunction
D. Modus Tollens
D. Modus Tollens (If p → q and ¬q, then ¬p)
30
30. What is a valid argument in logic?
A. An argument where all premises are true
B. An argument where the conclusion is true
C. An argument where if all premises are true, the conclusion must be true D. An argument where some premises are false
C. An argument where if all premises are true, the conclusion must be true (Valid argument)
31
31. What is the difference between inclusive and exclusive or? A. Inclusive or allows both statements to be true, exclusive doesn't B. Inclusive or requires both statements to be true
C. Exclusive or allows both statements to be true
D. There is no difference
A. Inclusive or allows both statements to be true, exclusive doesn’t (Difference between OR types)
32
32. What is a sufficient condition in a conditional statement?
A. The consequent
B. The antecedent
C. Both antecedent and consequent
D. Neither antecedent nor consequent
B. The antecedent (Sufficient condition is the if-clause)
33
33. What is the law of detachment?
A. Another name for Modus Tollens B. Another name for Modus Ponens
C. Another name for Hypothetical Syllogism
D. Another name for Disjunctive Syllogism
B. Another name for Modus Ponens (Law of Detachment)
34
34. What is a compound statement?
A. A statement with multiple subjects
B. A statement formed by combining two or more simple statements with logical connectives
C. A statement with multiple predicates
D. A statement that can't be broken down
B. A statement formed by combining two or more simple statements with logical connectives (Compound statement)
35
35. What is the purpose of a truth table?
A. To prove a statement is true
B. To show all possible combinations of truth values
C. To demonstrate logical fallacies
D. To identify valid arguments
B. To show all possible combinations of truth values (Purpose of truth table)
36
36. Which of the following statements is logically equivalent to "If p, then q"?
A. Not p or q
B. p and not q
C. Not p and q
D. p or not q
A. Not p or q (¬p ∨ q is equivalent to p → q)
37
37. In propositional logic, what is the contrapositive of "If p, then q"?
A. If q, then p
B. If not q, then not p
C. If not p, then not q
D. If p, then not q
B. If not q, then not p (Contrapositive)
38
38. Two compound statements P and Q are logically equivalent if and only if:
A. They have different truth values
B. Their truth table values are identical
C. They have opposite truth values
D. One is true when the other is false
B. Their truth table values are identical (Definition of logical equivalence)
39
39. The negation of "p if and only if q" is logically equivalent to:
A. p and not q
B. not p and q
C. p if q
D. p xor q (exclusive or)
D. p xor q (exclusive or) (Negation of biconditional)
40
40. Which of the following is NOT logically equivalent to "not(p and q)"?
A. Not p or not q
B. Not p and not q
C. Neither p nor q
D. Not p implies not q
B. Not p and not q (¬(p ∧ q) is De Morgan’s Theorem: ¬p ∨ ¬q, so ¬p ∧ ¬q is incorrect)
