Logic Practice 2018 [95 marks]

Transcription

1 Logic Practice 2018 [95 marks] Consider the following logic propositions. p: Sandi gets up before eight o clock q: Sandi goes for a run r: Sandi goes for a swim 1a. Write down in words the compound proposition If Sandi gets up before eight o clock then Sandi (either) goes for a run or goes for a swim, but not both. (A1)(A1)(A1) (C3) Note: Award (A1) for If then, (A1) for all propositions in the correct order, (A1) for or but not both (do not accept either as a replacement for but not both ). 1b. Complete the following truth table.

2 (A1)(A1)(ft) (C2) Note: Award (A1) for correct (q r) column, and (A1)(ft) for their correct p (q r) column. Follow through from their (q r) column. 1c. On a morning when Sandi does not get up before eight o clock, use your truth table to determine whether p (q r) is a tautology, contradiction or neither. tautology (A1)(ft) (C1) Note: Follow through from part (b). 2a. Consider the following statements z : x is an integer q : x is a rational number r : x is a real number. i) Write down, in words, q. ii) Write down a value for x such that the statement q is true. i) x is not a rational number (A1) Note: Accept x is an irrational number. ii) any non-rational number (for example: π, 2, ) (A1) (C2) 2b. Write the following argument in symbolic form: If is a real number and is not a rational number, then is not an integer. x x x

3 (r q) z (A1)(A1)(A1) (C3) Note: Award (A1) for seen, (A1) for z as the consequent and (A1) for (r q) or ( q r) as the antecedent (the parentheses are required). 2c. Phoebe states that the argument in part (b) can be shown to be valid, without the need of a truth table. Justify Phoebe s statement. all integers are rational numbers (and therefore x cannot be an integer if it is not a rational number) (R1) Note: Accept equivalent expressions. OR if x is an integer, then x is a rational number, therefore if x is not a rational number, then x is not an integer (contrapositive) (R1) (C1) Note: Accept If x is not in Q, then x is not in Z with a Venn diagram showing R, Q and Z correctly. 3a. Consider the following propositions: p : The lesson is cancelled q : The teacher is absent r : The students are in the library. Write, in words, the compound proposition q (p r). if the teacher is absent then the lesson is cancelled and the students are in the library (A1)(A1)(A1) (C3) Note: Award (A1) for If then. For Spanish candidates, only accept Si and entonces. For French candidates, only accept Si and alors. For all three languages these words are from the subject guide. Award (A1) for and, Award (A1) for correct propositions in correct order. Complete the following truth table. 3b.

4 (A1)(A1)(ft) (C2) Note: Award (A1) for r column correct and (A1) for q r column correct. Award (A0)(A1)(ft) for a q r column that correctly follows from an incorrect r column. 3c. Hence, justify why q r is not a tautology. not all of the entries are true (or equivalent) (R1) (C1) Note: Accept One entry is false. p : x is a multiple of 12 q : x is a multiple of 6. 4a. Write down in words p. x is not a multiple of 12 (A1) (C1) 4b. Write down in symbolic form the compound statement r : If x is a multiple of 12, then x is a multiple of 6. p q (A1)(A1)(C2) Note: Award (A1) for, (A1) for p and q in the correct order. Accept q p. 4c. Consider the compound statement s : If x is a multiple of 6, then x is a multiple of 12. Identify whether s : is the inverse, the converse or the contrapositive of r.

5 Converse (A1) (C1) 4d. Consider the compound statement s : If x is a multiple of 6, then x is a multiple of 12. Determine the validity of s. Justify your decision. not valid (A1) for example 18 is a multiple of 6 and not a multiple of 12 (R1) (C2) Notes: Do not award (A1)(R0). Any multiple of 6 that is not a multiple of 12 can be accepted as a counterexample. Consider the propositions r, p and q. 5a. Complete the following truth table. [4 marks] (A1)(A1)(ft)(A1)(ft)(A1) (C4) Notes: Award (A1) for each correct column. For the (r p) q follow through from the r p column. For the ((r p) q)) column, follow through from the preceding column.

6 5b. Determine whether the compound proposition ((r p) q)) (r p) q is a tautology, a contradiction or neither. Give a reason. tautology (A1)(ft) columns ((r p) q)) and (r p) q are identical (R1)(C2) Notes: Do not award (R0)(A1)(ft). Follow through from their table in part (a). Award the (R1) for an additional column representing ((r p) q)) (r p) q that is consistent with their table. Consider the following statements. p: the land has been purchased q: the building permit has been obtained r: the land can be used for residential purposes 6a. Write the following argument in symbolic form. If the land has been purchased and the building permit has been obtained, then the land can be used for residential purposes. (p q) r (A1)(A1)(A1) Notes: Award (A1) for conjunction seen, award (A1) for implication seen, award (A1) for correct simple propositions in correct order (the parentheses are required). Accept r (p q). In your answer booklet, copy and complete a truth table for the argument in part (a). 6b. Begin your truth table as follows.

7 (A1)(ft)(A1)(ft) Notes: Award (A1)(ft) for each correct column, follow through to the final column from their (p q) column. For the second (A1)(ft) to be awarded there must be an implication in part (a). Follow through from part (a). 6c. Use your truth table to determine whether the argument in part (a) is valid. Give a reason for your decision. The argument is not valid since not all entries in the final column are T. (A1)(ft)(R1) Notes: Do not award (A1)(ft)(R0). Follow through from part (b). Accept The argument is not valid since (p q) r is not a tautology. 6d. Write down the inverse of the argument in part (a) (i) in symbolic form; [4 marks] (ii) in words. (i) (p q) r (A1)(ft)(A1)(ft) OR ( p q) r (A1)(ft)(A1)(ft) Notes: Award (A1)(ft) for the negation of their antecedent and the negation of their consequent, (A1)(ft) for their fully correct answer. Follow through from part (a). Accept r (p q) or r ( p q). Follow through from part (a). (ii) if it is not the case that the land has been purchased and the building permit has been obtained then the land can not be used for residential purposes. (A1)(A1)(ft) OR if (either) the land has not been purchased or the building permit has not been obtained then the land can not be used for residential purposes. (A1)(A1)(ft) Notes: Award (A1) for if then seen, (A1)(ft) for correct statements in correct order. Follow through from part (d)(i).

8 Consider the statement p q. If I break my arm, then it will hurt. 7a. Write down in words, the inverse of p q. If I do not break my arm, then it will not hurt (A1)(A1) (C2) Note: Award (A1) for if then For Spanish candidates, only accept Si and entonces. Award (A1) for not break my arm and not hurt in correct order. 7b. Complete the following truth table. (A1)(A1) (C2) Notes: Award (A1) for each correct column. 7c. State whether the converse and the inverse of an implication are logically equivalent. Justify your answer. logically equivalent (A1)(ft) last two columns of the truth table are identical (R1)(ft) (C2) Notes: Do not award (A1)(ft)(R0). Follow through from the last two columns of the table in part (a).

9 Tuti has the following polygons to classify: rectangle (R), rhombus (H), isosceles triangle (I), regular pentagon (P), and scalene triangle (T). In the Venn diagram below, set A consists of the polygons that have at least one pair of parallel sides, and set B consists of the polygons that have at least one pair of equal sides. 8a. Complete the Venn diagram by placing the letter corresponding to each polygon in the appropriate region. For example, R has already been placed, and represents the rectangle. (A3) (C3) Note: Award (A3) if all four letters placed correctly, (A2) if three letters are placed correctly, (A1) if two letters are placed correctly. 8b. State which polygons from Tuti s list are elements of (i) A B; (ii) (A B). (i) Rhombus and rectangle OR H and R (A1)(ft) (ii) Scalene triangle OR T (A2)(ft) (C3) Notes: Award (A1) for a list R, H, I, P seen (identifying the union). Follow through from their part (a). Two propositions are defined as follows: p : Quadrilateral ABCD has two diagonals that are equal in length. q : Quadrilateral ABCD is a rectangle. 9a. Express the following in symbolic form. A rectangle always has two diagonals that are equal in length.

10 q p (A1)(A1) (C2) Note: Award the first (A1) for seeing the implication sign, the second (A1) is for a correct answer only. Not using the implication earns no marks. 9b. Write down in symbolic form the converse of the statement in (a). p q (A1)(ft) (C1) Note: Award (A1)(ft) where the propositions in the implication in part (a) are exchanged. 9c. Determine, without using a truth table, whether the statements in (a) and (b) are logically equivalent. Not equivalent; a kite or an isosceles trapezium (for example) can have diagonals that are equal in length. (A1)(R1) (C2) Notes: Accept a valid sketch as reasoning. If the reason given is that a square has diagonals of equal length, but is not a rectangle, then award (R1)(A0). Do not award (A1)(R0). Do not accept solutions based on truth tables. 9d. Write down the name of the statement that is logically equivalent to the converse. Inverse (A1) (C1) Note: Do not accept symbolic notation.

11 Consider the three propositions p, q and r. p: The food is well cooked q: The drinks are chilled r: Dinner is spoilt 10a. Write the following compound proposition in words. (p q) r If the food is well cooked and the drinks are chilled then dinner is not spoilt. (A1)(A1)(A1) (C3) Note: Award (A1) for If then (then must be seen), (A1) for the two correct propositions connected with and, (A1) for not spoilt. Only award the final (A1) if correct statements are given in the correct order. 10b. Complete the following truth table. (A1)(A1)(A1)(ft) (C3) Notes: Award (A1) for each correct column. The final column must follow through from the previous two columns.

12 Two propositions p and q are defined as follows p: Eva is on a diet q: Eva is losing weight. 11a. Write down the following statement in words. q p If Eva is losing weight then Eva is on a diet (A1)(A1) (C2) Notes: Award (A1) for If then For Spanish candidates, only accept Si and entonces. For French candidates, only accept Si and alors. For all 3 languages these words are from the subject guide. Award (A1) for correct propositions in correct order. 11b. Write down, in words, the contrapositive statement of q p. If Eva is not on a diet then she is not losing weight (A1)(A1) (C2) Notes: Award (A1) for not on a diet and not losing weight seen, (A1) for complete correct answer. No follow through from part (a). 11c. Determine whether your statement in part (a) is logically equivalent to your statement in part (b). Justify your answer. The statements are logically equivalent (A1)(ft) The contrapositive is always logically equivalent to the original statement OR A correct truth table showing the equivalence (R1)(ft) (C2) (R1)(ft) Note: Follow through from their answers to part (a) and part (b).

13 Consider the propositions p: I have a bowl of soup. q: I have an ice cream. 12a. Write down, in words, the compound proposition p q. If I do not have a bowl of soup then I have an ice cream. (A1)(A1) (C2) Notes: Award (A1) for If then Award (A1) for correct statements in correct order. 12b. Complete the truth table. (A1)(A1)(ft) (C2) Note: Follow through from third column to fourth column. 12c. Write down, in symbolic form, the converse of p q.

14 q p (A1)(A1) (C2) Notes: Award (A1) for. Award (A1) for q and p in correct order. Accept p q. Consider the following logic propositions: p : Yuiko is studying French. q : Yuiko is studying Chinese. 13a. Write down the following compound propositions in symbolic form. (i) Yuiko is studying French but not Chinese. (ii) Yuiko is studying French or Chinese, but not both. (i) p q (A1)(A1) Note: Award (A1) for conjunction, (A1) for negation of q. (ii) p q OR (p q) (p q) (A1) (C3) 13b. Write down in words the inverse of the following compound proposition. If Yuiko is studying Chinese, then she is not studying French. If Yuiko is not studying Chinese, (then) she is studying French. (A1)(A1)(A1) (C3) Notes: Award (A1) for if (then) seen, award (A1) for not studying Chinese seen, (A1) for correct propositions in correct order.

15 14a. Complete the truth table. (A1) for third column and (A1)(ft) for fourth column (A1)(A1)(ft) (C2) 14b. Consider the propositions p and q: p: x is a number less than 10. q: x2 is a number greater than 100. Write in words the compound proposition p q. x is greater than or equal to (not less than) 10 or x 2 is greater than 100. (A1)(A1) (C2) Note: Award (A1) for greater than or equal to (not less than) 10, (A1) for or x 2 is greater than c. Using part (a), determine whether p q is true or false, for the case where x is a number less than 10 and x 2 is a number greater than 100. True (A1)(ft) (C1) Note: Follow through from their answer to part (a). 14d. Write down a value of x for which is false. p q

16 Any value of x such that 10 x < 10. (A1)(ft) (C1) Note: Follow through from their answer to part (a). Consider the following propositions. p : Students stay up late. q : Students fall asleep in class. 15a. Write the following compound proposition in symbolic form. If students do not stay up late then they will not fall asleep in class. p q (A1)(A1) (C2) Note: Award (A1) for any 2 correct symbols seen in a statement, (A1) for all 3 correct symbols in correct order. 15b. Complete the following truth table. (A1)(A1)(ft)(A1)(ft) (C3) Note: Award (A1) for each correct column. 4 th column is follow through from 3 rd, 5 th column is follow through from 4 th. 15c. Write down a reason why the statement (p q) is not a contradiction.

17 Not all of last column is F (R1)(ft) (C1) Note: Award (R1)(ft) if final column does not lead to a contradiction. International Baccalaureate Organization 2018 International Baccalaureate - Baccalauréat International - Bachillerato Internacional Printed for North hills Preparatory