IB Mathematical Studies Logic Practice Name Note the markscheme is at the end!!!! 1. Police in a town are investigating the theft of mobile phones one evening from three cafés, Alan s Diner, Sarah s Snackbar and Pete s Eats. They interviewed two suspects, Matthew and Anna about that evening. Matthew said: I visited Pete s Eats and visited Alan s Diner and I did not visit Sarah s Snackbar Let p, q and r be the statements: p : I visited Alan s Diner q : I visited Sarah s Snackbar r : I visited Pete s Eats (a) Write down Matthew s statement in symbolic logic form. What Anna said was lost by the police, but in symbolic form it was (q r ) p Write down, in words, what Anna said. 2. In a particular school, students must choose at least one of three optional subjects: art, psychology or history. Consider the following propositions a: I choose art, p: I choose psychology, h: I choose history. (a) Write, in words, the compound proposition h ( p a). IB Questionbank Mathematical Studies 3rd edition 1
Complete the truth table for a p. a p a a p T T F T F F F T T F F T (1) State whether a p is a tautology, a contradiction or neither. Justify your answer. 3. Consider two propositions p and q. (a) Complete the truth table below. p q q p q p p q T T T F F T F F (4) Decide whether the compound proposition (p q) ( p q) is a tautology. State the reason for your decision. 4. (a) Complete the truth table shown below. p q p q T T T F F T F F p (p q) (p ( p q)) p IB Questionbank Mathematical Studies 3rd edition 2
State whether the compound proposition(p ( p q)) p is a contradiction, a tautology or neither. (1) Consider the following propositions. p: Feng finishes his homework q: Feng goes to the football match Write in symbolic form the following proposition. If Feng does not go to the football match then Feng finishes his homework. 5. (a) Complete the following truth table. p q... p q T T F T F T F T F F F T Consider the propositions............ p: Cristina understands logic q: Cristina will do well on the logic test. Write down the following compound proposition in symbolic form. If Cristina understands logic then she will do well on the logic test Write down in words the contrapositive of the proposition given in part. 6. Consider the two propositions p and q. IB Questionbank Mathematical Studies 3rd edition 3
p: The sun is shining q: I will go swimming Write in words the compound propositions (a) p q; p q. The truth table for these compound propositions is given below. p q p q p p q T T T T T F F F F T T T F F T T Complete the column for p. (1) (d) State the relationship between the compound propositions p q and p q. (1) 7. Consider the following propositions p: The number is a multiple of five. q: The number is even. r: The number ends in zero. (a) Write in words (q r) p Consider the statement If the number is a multiple of five, and is not even then it will not end in zero. (i) (ii) Write this statement in symbolic form. Write the contrapositive of this statement in symbolic form. (6) (Total 9 marks) 8. Consider the statement p: IB Questionbank Mathematical Studies 3rd edition 4
If a quadrilateral is a square then the four sides of the quadrilateral are equal. (a) Write down the inverse of statement p in words. Write down the converse of statement p in words. Determine whether the converse of statement p is always true. Give an example to justify your answer. Consider the statement If the number is a multiple of five, and is not even then it will not end in zero. (i) (ii) Write this statement in symbolic form. Write the contrapositive of this statement in symbolic form. (6) (Total 9 marks) IB Questionbank Mathematical Studies 3rd edition 5
Worked Out Solutions 1. (a) r p q 1)(A1) Note: Award (A1) for two conjunctions, (A1) for negation seen on q, (A1) for correct compound statement. (C3) If I visited (either) Sarah s Snackbar or Pete s Eats (then) I did not visit Alan s Diner 1)(A1) Note: Award (A1) for If (then), (A1) for Sarah s Snackbar or Pete s Eats, (A1) for did not visit Alan s Diner. (C3) 2. (a) If I do not choose history then I choose either psychology or I choose art 1)(A1) Notes: Award (A1) for if (then) Award (A1) for not choose history. Award (A1) for choose (either) psychology or art (or both). If the order of the statements is wrong award at most 1)(A0) (C3) a p a a p T T F T T F F T F T T T F F T F (A1) (C1) Neither, because not all the entries in the last column are the same )(R1) Notes: Do not award (R0)(A1). Follow through from their answer to part. Reasoning must be consistent with their answer to part. (A1)(ft (C2) IB Questionbank Mathematical Studies 3rd edition 6
3. (a) p q q p q p p q T T F F F T T F T T F T F T F T T T F F T T T F 1)(ft) 1)(ft) Note: Award (A1) for each correct column (second column (ft) from first, fourth (ft) from third). Follow through from second column to fourth column for a consistent mistake in implication. (C4) Since second and fourth columns are not identical (R1)(ft) Not a tautology (A1)(ft ) (C2) Note: (R0)(A1) may not be awarded. 4. (a) p q p q p (p q) (p ( p q)) p T T T T T T F F T T F T F F T F F F F T 1)(ft)(A1)(ft) Note: Award (A1) for each correct column. (C3) tautology (A1)(ft ) (C1) Note: Follow through from their last column. q p IB Questionbank Mathematical Studies 3rd edition 7
Note: Award (A1) for q and p in correct order, (A1) for sign. 5. (a) p q q p q T T F F T F T T F T F T F F T T Note: Award (A1) for q, (A1) for last column. p q Note: Award (A1) for, (A1) for p and q in the correct order. If Cristina does not do well on the logic test then she does not understand logic. Note: Award (A1) for If (then), must be an implication, (A1) for the correct propositions in the correct order. 6. (a) If the sun is shining then I will go swimming. Note: Award (A1) for if then and (A1) for correct order. Either the sun is not shining or I will go swimming. Note: Award (A1) for both correct statements and (A1) for either or. p q p q p p q T T T F T T F F F F F T T T T F F T T T IB Questionbank Mathematical Studies 3rd edition 8
(A1) (C1) (d) They are (logically) equivalent. (A1) (C1) Note: Do not accept any other answers. 7. (a) If the number is even and the number does not end in zero, (then) the number is not a multiple of five. 1)(A1) Note: Award (A1) for if (then), (A1) for the number is even and the number does not end in zero, (A1) for the number is not a multiple of 5. (i) (p q) r 1)1) (A1) for, (A1) for, (A1) for p and q, (A1) for r Note: If parentheses not present award at most 1)0). (ii) r ( p q) OR r (p q) )(A1)(ft) Note: Award (A1)(ft) for reversing the order, (A1) for negating the statements on both sides. If parentheses not present award at most (A1)(ft)(A0). Do not penalise twice for missing parentheses in (i) and (ii). (A1)(ft [9] 8. (a) If a quadrilateral is not a square (then) the four sides of the quadrilateral are not equal. Note: Award (A1) for if (then), (A1) for the correct phrases in the correct order. If the four sides of the quadrilateral are equal (then) the quadrilateral is a square. 1)(ft) Note: Award (A1) for if (then), (A1)(ft) for the correct phrases in the correct order. (C2) Note: Follow through in if the inverse and converse in (a) and are correct and reversed. IB Questionbank Mathematical Studies 3rd edition 9
The converse is not always true, for example a rhombus (diamond) is a quadrilateral with four equal sides, but it is not a square. (A1)(R Note: Do not award (A1)(R0). IB Questionbank Mathematical Studies 3rd edition 10