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Math 108 UC Davis – Questions With Correct
Solutions
Proposition Correct Answer - A proposition is a declarative statement that is either true or false, but not both. Proposition Examples Correct Answer - f(x) = |x| is differentiable at x= f(x)= |x-3| / x-3 is a continuous function sqrt(3) is not rational Every continuous function is differentiable Any square matrix has an inverse All UCD students wear glasses det(A) = 0 implies that A has a Ax=0 has a non-zero solution Non-Proposition Statements Correct Answer - -Imperative (commands) -Interrogative (questions) -Exclamatory -Open Sentence -Paradox Open Sentence Correct Answer - Not a proposition, because values of x can make it true OR false Ex: ln(x^2+1) = 0 x+2 = 2x
Paradox Correct Answer - Is not a proposition Ex: This statement is false. Proposition Correct Answer - A proposition is a sentence that has exactly one true value. It is either True, which we denote by T, or False, which we denote by F. Negation Correct Answer - The negation of proposition P, denoted ~P, is the proposition "not P". The proposition ~P is True exactly when P is False. Conjunction Correct Answer - Given propositions P and Q, the conjunction of P and Q, denoted P ∧Q, is the proposition "P AND Q". P ∧Q is true exactly when BOTH P and Q are true. Truth Value Correct Answer - The form P ∧Q itself has no truth value. Only when the components P and Q are assigned to be specific propositions does P ∧Q have the value T or F. Disjunction Correct Answer - Given propositions P and Q, the disjunction of P and Q, denoted P ∨Q, is the proposition "P or Q". P ∨Q is true exactly when AT LEAST ONE of P or Q is true. Symbolize: "Either 7 is prime and 9 is even, or else 11 is not less than 3" Correct Answer - (P ∧Q) ∨~R P: "7 is prime" Q: "9 is even" R: "11 is less than 3" A proposition with 3 components (P, Q, R) has how many possible combinations of truth values in its truth table? Correct Answer - 2^3 possible combinations of truth values
Associative Laws Equivalent propositions (f)(g): Correct Answer - P ∧(Q ∨R) and (P ∧Q) ∨(P ∧R) P ∨(Q ∧R) and (P ∨Q) ∧(P ∨R) Distributive Laws Equivalent propositions (h)(i): Correct Answer - ~(P ∧Q) and ~P ∨~Q ~(P ∨Q) and ~P ∧~Q DeMorgan's Laws DENIAL of a proposition Correct Answer - A DENIAL of proposition P is any proposition equivalent to ~P. -A proposition only has one negation, ~P, but many denials such as ~~P, ~~~P, etc. Denials of "x is odd" include: "x is not odd" "x is even" "x is divisible by two" "Neither P nor Q" Correct Answer - ~(P ∨Q) "Either not P or else Q" Correct Answer - (~P) ∨Q Conditional Sentence Correct Answer - For propositions P and Q, the conditional sentence P => Q is the proposition "If P, then Q". Proposition P is called the ANTECEDENT Q is the CONSEQUENT
-The sentence P=>Q is true if and only if P is false or Q is true. P => Q Correct Answer - is true if and only if P is false or Q is true. The truth value of P=>Q depends only on the truth value of components P and Q, even when there is no connection to be made between the antecedent and the consequent Correct Answer - All of the following are TRUE: "if sine( 𝛑 )=1, then 6 is prime" (F, F, T) (line 4 of truth table) "13 > 7 => 2+3=5" (T, T, T) (line 1 of truth table) " 𝛑 = 3 => Paris is the capital of France." (F, T, T) (line 2 of truth table) Both of these sentences are false by line 3 of the P=>Q truth table... Correct Answer - "if Saturn has rings, then (2+3)^2 = 2^2 + 3^2." (T, F, F) "If 4 𝛑 > 10, then 1 is a prime number" (T, F, F) line 3 of truth table = T, F, F Two propositions associated with P => Q ... Correct Answer - its CONVERSE: Q => P and its CONTRAPOSITIVE: (~Q) => (~P) CONVERSE of P=>Q Correct Answer - Q => P CONTRAPOSITIVE of P=>Q Correct Answer - (~Q)=>(~P) Bi-Conditional Connective Correct Answer - For propositions P and Q, the biconditional sentence P<=>Q is the proposition "P IF AND
Universes Correct Answer - N, Z, Q, R, and C Truth Set for Universes N, Z, and R Correct Answer - N = {1, 2} Z = {-2, -1, 0, 1, 2} R = open interval (-sqrt(5), sqrt(5)) With a Universe specified... Correct Answer - two open sentences P(x) and Q(x) are equivalent if they have the same truth set. The Symbol: ∃ Correct Answer - The symbol ∃is called the EXISTENTIAL quantifier. For an open sentence P(x), the sentence ( x)P(x) is read as "there∃ EXISTS x SUCH THAT P(x)", or "For some x, P(x)". The sentence ( x)P(x) is true if the truth set of P(x) is non-empty.∃ The Symbol: ∀ Correct Answer - The symbol ∀is called the UNIVERSAL quantifier. For an open sentence P(x), the sentence ( ∀x)P(x) is read "FOR ALL x, P(x)", or "For every x, P(x)". The sentence ( ∀x)P(x) is true if the truth set of P(x) is the ENTIRE Universe. Universally Quantified sentences that are TRUE in Universe R: Correct Answer - 1. ( ∀x)(x<0 or x=0 or x>0)
- ( ∀x)(2^x > 0)
- ( ∀x)(x+2 > x) Words that require Universal Quantifiers: Correct Answer - "for all"
"for every" "for each" Words that require Existential Quantifiers: Correct Answer - "some" "at least one" "there exist(s)" "there is (are)" "All apples have spots", when Universe= just apples Correct Answer - ( ∀x)(x has spots) "All apples have spots", when Universe= all fruits Correct Answer - A(x): x is an apple S(x): x has spots ( ∀x) [A(x) => S(x)] : "For all x in the universe, if x is an apple, then x has spots". Symbolic translation: "Some apples have spots" Correct Answer - A(x): x is an apple S(x): x has spots ( x) [A(x) ^ S(x)] :∃ "There is an object x such that it is an apple and it has spots." "All P(X) are Q(x)" should be symbolized... Correct Answer - ( ∀x) [P(x) => Q(x)] "Some P(x) are Q(x)" should be symbolized... Correct Answer - ( x) [P(x) ^ Q(x)]∃ Translate the sentence using quantifiers: "For every odd prime x less than 10, x^2+4 is prime" Correct Answer
- ( ∀x) (x is prime ∧ x is odd ∧x<10 => x^2+4 is prime)
If A(x) is an open sentence with variable x, then... Correct Answer - (a) ( !x)A(x) => ( x)A(x)∃ ∃ (b) ( !x)A(x) is equivalent to ( x)A(x)∃ ∃ ∧ ( ∀y)( ∀z)[A(y) ∧A(z) => y=z] THEOREM definition Correct Answer - A THEOREM in mathematics is a statement that describes a pattern or relationship among quantities or structures. PROOF definition Correct Answer - A PROOF of a theorem is a justification of the truth of the theorem that follows the principles of logic AXIOMS (postulates) Correct Answer - AXIOMS (or POSTULATES) are an initial set of statements that are assumed to be true. *Theorems are true in any situation where their axioms are true UNDEFINED TERMS Correct Answer - Concepts that are fundamental to the context of the study.
- In any proof at any time you may... Correct Answer - State an axiom, an assumption, or a previously proved result. A result that serves as a preliminary step is often called a... Correct Answer - Lemma
- In any proof at any time you may use the tautology rule: Correct Answer - State a sentence whose symbolic translation is a tautology.
- In any proof at any time you may use the replacement rule: Correct Answer - State a sentence equivalent to any statement earlier in the proof.
***A thorough knowledge of the logical equivalences of Theorems 1.1. and 1.2.2 is essential when one uses the REPLACEMENT RULE bc these replacements are done routinely.
- In any proof at any time you may: Correct Answer - Use a definition to state an equivalent to a statement earlier in the proof. Example: Divisibility is defined as it is, so that in a proof we can replace the statement "a divides b" with "b= a ∙ k for some integer k". The most fundamental rule of reasoning is MODUS PONENS, which is... Correct Answer - Modus Ponens is based on the tautology: P ∧[P => Q] => Q -This means that whenever P and P=>Q are both true, we may deduce that Q is also true. -This rule allows us to make a connection so that we can get from a statement P to a different statement Q
- In any proof at any time you may use the Modus Ponens rule: Correct Answer - After statement P and P => Q appear in a proof, state Q. A DIRECT PROOF of a statement of the form P=>Q... Correct Answer - proceeds in a step-by-step fashion from the antecedent P to the consequent Q -Since P=>Q is false only when P is true and Q is false, it is sufficient to show that this situation cannot happen. Strategy for developing a Direct Proof of a Conditional Sentence: Correct Answer - 1. Determine the hypothesis(if any) and the antecedent and consequent
Method of Contradiction Correct Answer - An indirect proof of a statement P by the METHOD OF CONTRADICTION uses the logic that if P can not be false, then P must be true. -To use this method we temporarily assume that P is false and then see what would happen. -If what happens is a contradiction, then we know that P must be true. Method of Contradiction symbols: Correct Answer - If Q is any proposition, then [(~P) => (Q ∧~Q)] is equivalent to P. Method of Contradiction FORM: Correct Answer - Proof: Suppose ~P. . . Therefore, Q. . . Therefore, ~Q. Hence, Q ∧~Q is a contradiction Thus, P. Proofs of BI-CONDITIONAL sentences often make use of the fact... Correct Answer - that P <=> Q is equivalent to P=>Q ∧Q=>P Squares of even numbers... Correct Answer - are always even Squares of odd numbers... Correct Answer - are always odd The square root of an even number... Correct Answer - is always even The square root of an odd number... Correct Answer - is always odd
Two times any number x... Correct Answer - is always an even number Even x Even Correct Answer - Even Even x Odd Correct Answer - Even Odd x Even Correct Answer - Even Odd x Odd Correct Answer - Odd Even + Even Correct Answer - Even Even + Odd Correct Answer - Odd Odd + Even Correct Answer - Odd Odd + Odd Correct Answer - Even Set Notation Correct Answer - {x: P(x)} The Set A = {1,3,5,7,9,11} may be written as... Correct Answer - {x: x ∈N, x is odd, and x<14} R Correct Answer - Set of all Real Numbers N Correct Answer - Set of all Natural Numbers Z Correct Answer - Set of all Integers Q Correct Answer - Set of all Rational Numbers R-Q Correct Answer - Set of all Irrational Numbers
