Math Assignment 2 Solutions - Spring Jaimos F Skriletz Provide definitions for the following:

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1 Math Assignment 2 Solutions - Spring Jaimos F Skriletz 1 1. Provide definitions for the following: (a) A statement is a declarative sentence that is either true or false, but not both at the same time. (b) If P and Q are two statements then the conjunction (logical and ) is the compound statement P Q (P and Q). The truth of the conjunction is defined by the following truth table. P Q P Q F T F (c) If P and Q are two statements then the disjunction (logical or ) is the compound statement P Q (P or Q). The truth of the disjunction is defined by the following truth table. P Q P Q T F T (d) If P is a statement then the negation of P (logical not ) is the statement P (not P). The truth of the negation is defined by the following truth table. P P T F F T (e) A tautology is a compound statement that is always true for all possible combinations of the truth s of its components. (f) Two compound statements P and Q are equivalent, written as P Q, if they produce the same truth in all possible combinations of the truth s of their components (same truth table). (g) If P and Q are two statements then the conditional (logical implication or if...then ) is the compound statement P Q (P implies Q). The truth of the conditional is defined by the following truth table. P Q P Q F F T (h) The converse of the conditional statement P Q is the conditional statement Q P. (i) The inverse of the conditional statement P Q is the conditional statement P Q. (j) The contrapositive of the conditional statement P Q is the conditional statement Q P. The contrapositive is equivalent to the original conditional statement, P Q Q P. (k) If P and Q are statements then the biconditional, written as P Q (P if and only if Q), is the compound statement (P Q) (Q P). (l) An argument consists of a set of statements called premises and another statement called the conclusion. An argument is valid if the conclusion is true when ever all the premises are true. (m) An argument is invalid if it is not a valid argument.

2 Math Assignment 2 Solutions - Spring Jaimos F Skriletz 2 2. There are various higher level logical operators which can be written in terms of the basic logical operators ( and, or, not ). Three examples of these are the Conditional (P Q), Xor (P Q) and Biconditional (P Q). I have provided a truth table for each of the logical operators, use this to write each one in terms of the basic logical operators. (a) The conditional statement P Q is equivalent to the compound statement P Q. The equivalency is shown by looking at their two truth tables P Q P Q F F T P Q P P Q T T F T T T F Since the two truth tables are identical the statements are equivalent, thus P Q P Q. (b) The xor statement is equivalent to the compound statement (P Q) ( P Q). The equivalency is shown by looking at their two truth tables P Q P Q T T F T F T P Q P Q P Q (P Q) ( P Q) T T T F T F T F T F F Since the two truth tables are identical the statements are equivalent, thus P Q (P Q) ( P Q). (c) The biconditional is defined to be the statement P Q (P Q) (Q P). In part (a) we showed that P Q P Q, thus we can write the biconditional as P Q (P Q) (Q P) ( P Q) ( Q P) 3. Let P, Q, R and S be statements. Create truth tables for the following compound statements. (a) P (Q P) P Q P (Q P) T T F (b) [R ( P Q)] (R Q) P Q R [R ( P Q)] (R Q) T T T F T F T T F F F (c) [(P Q) Q] R P Q R [(P Q) Q] R F T T F T F T F F T F T F F T F T

3 Math Assignment 2 Solutions - Spring Jaimos F Skriletz 3 (d) S [ ( R Q) P] P Q R S S [ ( R Q) P] T F F F T T F T F T T T T T F T F T F T T F F F T F T F T F F F F F F 4. Determine the validity of the following arguments: No wizard can yodel. (a) All lizards can yodel. No wizard is a lizard. By looking at an Euler Diagram for this argument we see that the two premises split the wizards and lizards as shown. Thus the conclusion is valid. Wizards things that yodel Lizards All prime numbers are odd. (b) 2 is a prime number. 2 is an odd number. By looking at the Euler Diagram for this argument the premises force 2 to be in the set of odd numbers. Thus this argument is valid. Note that an argument is valid is only based on the truth of the premises. In number theory this argument would not prove that 2 is odd because the premise all prime numbers are odd is not true. This is example is to illustrate that the validity of the argument is based on the promises and conclusion given and not its truth in some other accepted system. odd numbers prime numbers 2 This type of valid argument is common in mathematics. In general this kinda of argument is part of what they call proof by contradiction. If a valid argument leads to a false conclusion in your system the only possibility is that at least one of the premises is false. In this case this valid argument can be used to prove the statement All prime numbers are odd is false. And the reason this is a false statement is because the prime number 2 is not odd.

4 Math Assignment 2 Solutions - Spring Jaimos F Skriletz 4 If I get an A on the math exam, I will be happy. (c) I am happy. I got an A on the math exam. This argument is best analyzed by looking at it in symbolic form. We have two statements, P If I get an A and Q I am happy. In symbolic form this argument is: P Q Q P This is an invalid argument. The reason being is P Q only forces Q to be true if P is true. On the other hand if Q is true there is no way to know if P is true or false (it could be either). For example you could have not got an A on the exam but bought a wining loto ticket on the way home, so your happiness is not dependent on getting an A on the exam. This is a classic example of an invalid argument that depending on the choice language is commonly mistaken to be valid. Some whales make good pets. Some good pets are cute. (d) Some cute pets bite. Some whales bite. By looking at the Euler Diagram for this argument we see that there is not enough info to force the conclusion to be true. All though it is possible that some whales bite, it is not forced to be true by the premises. According to the promises it could also be possible that no whales bite, so the argument is invalid as the truth is not guaranteed. good pets whales things that bite cute things 5. Provide a valid conclusion to the following arguments: (a) The Euler Diagram for this argument is as shown. All Ruben sandwiches are good. All good sandwiches have pastrami. All sandwiches with pastrami need mustard. Things that need mustard Things that have pastrami Things that are good Ruben Sandwitches Thus a valid logical conclusion to this argument is that All Ruben sandwiches need mustard.

5 Math Assignment 2 Solutions - Spring Jaimos F Skriletz 5 (b) All multiples of 11 end with a is a multiple of 11. The Euler Diagram for this argument is as shown. Numbers that end in 5 Multiples of Thus the valid conclusion to the argument is that 1001 ends with a 5. Note that a valid conclusion to an argument is only based on the premises used. In this case the premise All multiples of 11 end with a 5 is not true in the language of number theory. This is another example to point out what makes a valid argument in such a way that you don t relay on your accepted truths when finding logical conclusions. 6. Consider the following Lewis Carroll syllogism. In the universe of kittens come to a valid conclusions of the statements: (a) No kitten, that loves fish, is unteachable. (b) No kitten without a tail will play with a gorilla. (c) Kittens with whiskers always love fish. (d) No teachable kitten has green eyes. (e) No kittens have tails unless they have whiskers. The first step is to put the given information into symbolic form. Use the following statement definitions (a) A Kittens that love fish. (b) B Kittens that are teachable. (c) C Kittens that have a tail. (d) D Kittens that play with gorillas. (e) E Kittens with whiskers. (f) F Kittens with green eyes. With these definitions the given statements in symbolic notation are given below. Note that the contrapositive is equivalent to the conditional, so for each I will provide the conditional and its contrapositive as equivalent statements. (a) A B B A (b) C D D C (c) E A A E (d) B F F B (e) C E E C In the above the statements D and F are only mentioned once so it may be possible to put them into a chain. Putting all the statements above together we generate the chain D C E A B F This says the conclusion to the syllogism is D F or F D (contrapositives). The final conclusion written in the language of the problem is the statement F D which states that Kittens with green eyes do not play with gorillas.

6 Math Assignment 2 Solutions - Spring Jaimos F Skriletz 6 7. Provide a valid logical argument to prove that A B A B using the basic definitions of set theory. The three basic definitions needed from set theory for this proof are (a) Subset: S T means that if x S then x T. (b) Intersection: A B = {x : x A and x B}. (c) Union: A B = {x : x A or x B}. A proof of A B A B goes as follows. Let x A B then x A and x B. If x A and x B then x A or x B. Therefor x A B. Thus I have shown that for every x A B then x A B. So by definition of subset, A B A B. Note that the validity of this argument rests upon the line If x A and x B then x A or x B. In symbolic logic this is the statement (P Q) (P Q). You can check that the statement is a tautology and thus a valid argument.