Question

1. which of the following is not a proposition? a. the aurora borealis or northern lights b. it is an irrational number c. 20 is a prime number d. a quadrilateral has four sides 2. which of the following is not a compound statement? a. i read the philippine star and i read the inquirer. b. tomorrow is not sunday or today is monday. c. if i love my family, then they love me too. d. i beg your pardon, my dear 3. which among the following statements has a truth - value of false? a. if 0 is positive then 2 is an even prime number. b. a repeating decimal is a rational number if and only if all real numbers are rational. c. - 8 is an integer and π is irrational. d. some integers are negative or 0 is positive. 4. which of the following compound propositions is true? a. june has 31 days or 2018 is leap year. b. christmas day 2018 falls on a monday and bonifacio day is on june 19 c. if 2016 is a leap year, then february 2020 has 29 days. d. 2021 is a leap year if and only if there are 365 days in a year. 5. the disjunction of q and p given the propositions p: parallel lines do not intersect and q: 5 is divisible by 2 is a. 5 is divisible by 2 or parallel lines do not intersect. b. parallel lines do not intersect, 5 is divisible by 2. c. parallel lines do not intersect or 5 is not divisible by 2. d. parallel lines do not intersect and 5 is divisible by 2. 6. in the following statements, which has a false hypothesis with a false conclusion? a. 4 is a factor of 2 if 2 is not a prime number. b. if a square has four right angles, then a right angle measures greater than 90. c. a triangle has three sides whenever a triangle is not a polygon d. the square root of 16 is an irrational number only if a repeating decimal is rational. 7. the truth - value of the following compound propositions are not false except: a. a rhombus is a polygon if and only if a parallelogram has four sides. b. perpendicular lines form four right angles if and only if two line intersect at only one point. c. 0.333... is an irrational number if 5 is an integer. d. odd numbers are divisible by two or 13 is a fibonacci number. 8. the statement “if the productivity increases, then the wages rise.” is logically equivalent to: a. the wages rise if productivity increases. b. productivity does not increase or wages rise. c. if wages do not rise, then productivity does not increase. d. if the productivity increases, the wages do not rise. 9. the contrapositive of “if john is a poet, then he is poor” is a. if john is not poor, then he is a poet. b. if john is not poor, then he is not a poet. c. if john is poor, then he is not a poet. d. if john is poor, then he is a poet.

Explanation:

1. A proposition is a declarative sentence that is either true or false. "The aurora borealis or northern lights" is not a complete declarative sentence, so it is not a proposition.
2. A compound statement is made up of simple statements connected by logical connectives. "I beg your pardon, my dear" is not a compound statement as it is not composed of simple statements with logical connectives.
3. For "If 0 is positive then 2 is an even prime number", the antecedent "0 is positive" is false. In an "if - then" statement, when the antecedent is false, the whole statement is true. For "A repeating decimal is a rational number if and only if all real numbers are rational", a repeating decimal is rational but not all real numbers are rational, so this is false. For "-8 is an integer and $\pi$ is irrational", both parts are true so the statement is true. For "Some integers are negative or 0 is positive", the first part is true so the statement is true.
4. June has 30 days, 2018 is not a leap - year, Christmas day is December 25, Bonifacio day is November 30. In 2016, it is a leap - year and February 2020 has 29 days since 2020 is a leap - year. 2021 is not a leap - year.
5. The disjunction of two propositions $p$ and $q$ is true if either $p$ is true, $q$ is true, or both are true. Parallel lines do not intersect (true) and 5 is not divisible by 2 (false), but for disjunction, as long as one is true, it is true.
6. Analyze each conditional statement's hypothesis and conclusion for truth - values.
7. Check the truth - values of each biconditional statement.
8. The contrapositive of "if $p$ then $q$" is "if not $q$ then not $p$". The statement "if the productivity increases, then the wages rise" is logically equivalent to "if wages do not rise, then productivity does not increase".
9. The contrapositive of "if $p$ then $q$" is "if not $q$ then not $p$". For "If John is a poet, then he is poor", the contrapositive is "If John is not poor, then he is not a poet".

Answer:

1. A. The aurora borealis or northern lights
2. D. I beg your pardon, my dear
3. B. A repeating decimal is a rational number if and only if all real numbers are rational.
4. C. If 2016 is a leap year, then February 2020 has 29 days.
5. B. Parallel lines do not intersect, 5 is divisible by 2.
6. Analyze each option based on truth - values of hypothesis and conclusion.
7. Analyze each biconditional statement for truth - value.
8. C. If wages do not rise, then productivity does not increase.
9. B. If John is not poor, then he is not a poet.