Solutions to Homework I (1.1)
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1 Solutions to Homework I (1.1) Problem 1 Determine whether each of these compound propositions is satisable. a) (p q) ( p q) ( p q) b) (p q) (p q) ( p q) ( p q) c) (p q) ( p q) (a) p q p q p q p q p q (p q) ( p q) ( p q) T T F F T T F F T F F T T F T F F T T F F T T F F F T T T T T T From the last column we see that the proposition is satisable. September 11, / 10
2 Solutions to Homework I (1.2) Problem 1 Determine whether each of these compound propositions is satisable. a) (p q) ( p q) ( p q) b) (p q) (p q) ( p q) ( p q) c) (p q) ( p q) (a) The same result can be obtained using logically equivalent transformations: (p q) ( p q) ( p q) ((p ( p q)) ( q ( p q))) ( p q) ((p p) (p q)) ( q p) ( q q)) ( p q) ((p q)) ( p q)) ( p q) p q This shows that the proposition is true if both p and q are false. Thus, the proposition is satisable. September 11, / 10
3 Solutions to Homework I (1.3) Problem 1 Determine whether each of these compound propositions is satisable. a) (p q) ( p q) ( p q) b) (p q) (p q) ( p q) ( p q) c) (p q) ( p q) (b) Considering that (p q) (p q) ( p q) ( p q) ( p q) ( p q) ( p q) ( p q) we can compose the truth table as follows: ( p q) ( p q) (p q) (p q) p q p q p q p q p q p q Entire proposition T T F F T F T T F T F F T F T T T F F T T F T T T F F F F T T T T F T F Hence, this proposition is not satisable (it is the contradiction). September 11, / 10
4 Solutions to Homework I (1.4) Problem 1 Determine whether each of these compound propositions is satisable. a) (p q) ( p q) ( p q) b) (p q) (p q) ( p q) ( p q) c) (p q) ( p q) (c) The truth table as follows: Hence, this proposition is not satisable. p q (p q) ( p q) T T T F F T F F F T F T F F T F F T F F Step: September 11, / 10
5 Solutions to Homework I (2.1) Problem 2 Use the laws in the lecture slide #28 to simplify the formula (p q) ( p q). Check the results using the technique of truth tables. (p q) ( p q) p q ( p q) De Morgan law p (( q p) ( q q)) associative and distributive laws p (( q p) T) complement law p ( q p) identity law q p assiociative and idempotent laws p q Example 2 on slide #24 Alternative answers may be q p or q p. September 11, / 10
6 Solutions to Homework I (2.2) Problem 2 Use the laws in the lecture slide #28 to simplify the formula (p q) ( p q). Check the results using the technique of truth tables. To check the result one can compose the following truth table: p q (p q) ( p q) p q T T F T F F F T F F T F T F T F F T T T F T T F T T T T T F F F T F T T F T T T Step: September 11, / 10
7 Solutions to Homework I (3) Problem 3 Find CNF (Conjunctive Normal Form) and DNF (Disjunctive Normal Form) for the formula (p q) r p. (p q) r p [(p q) r p] [r p (p q)] def. of biconditional [( p q) r p] [r p ( p q)] Example 2 on slide #24 [ ( p q) (r p)] [ (r p) p q] Example 2 on slide #24 [(p q) (r p)] [ (r p) p q] De Morgan law [(p q) (r p)] ( r p p q) De Morgan law [(p q) (r p)] ( r T q) complement law [(p q) (r p)] T domination law (p q) (r p) domination law (p r) ( q r) ( q p) distributive law The last two lines yield DNF and CNF respectively: DNF: (p q) (r p) CNF: (p r) ( q r) ( p q) September 11, / 10
8 Solutions to Homework I (4.1) Problem 4 Show that and form a functionally complete collection of logical operators. Find a formula containing propositional variables p, q and r, that is true if and only if exactly one of those variables is true. Represent this formula in the form using only connectives and. Every proposition can be converted into DNF or CNF containing connectives, and. Using the De Morgan law, one can substitute every by its logical equivalent: p q ( p q). the proposition that is true i exactly one of those variables is true has three alternatives when it is valid: (a) only p is true: p q r T (b) only q is true: p q T (c) only r is true: p q r T So, the proposition can be written as (p q r) ( p q r) ( p q r). Using distributive and absorption laws several times, we see that this formula is logically equivalent to (p q r) ( q r) ( p r) ( p q) (1) Convert using the De Morgan law: ( p q r) (q r) (p r) (p q). September 11, / 10
9 Solutions to Homework I (4.2) Problem 4 Show that and form a functionally complete collection of logical operators. Find a formula containing propositional variables p, q and r, that is true if and only if exactly one of those variables is true. Represent this formula in the form using only connectives and. To verify the formula against the initial requirements, we construct the truth table of the proposition (1): p q r (p q r) ( q r) ( p r) ( p q) T T T T F F F F F F T T F T T T T T F F T F T T T T F F F T T F F T T T T T T T F T T T F F F T F T F T F T T T T T T T F F T T T T T T T T F F F F F T F T F T Step: September 11, / 10
10 Problems for Sept. 11, Let A = {1,2,3,4,5}. Determine the truth value of each of the following statements: (a) ( x A)(x + 3 = 10) (b) ( x A)(x + 3 < 10) (c) ( x A)(x + 3 < 5) (d) ( x A)(x + 3 7) 2 Determine the truth value of each of the following statements where U = {1,2,3} is the universal set: (a) x y. x 2 < y + 1 (b) x y. x 2 + y 2 < 12 (c) x y. x 2 + y 2 < 12 3 Negate each of the following statements: (a) x y. p(x, y) (b) x y. p(x, y) (c) y x z. p(x, y, z) 4 Negate each of the following statements: (a) All students live in the dormitories. (b) All mathematics majors are males. (c) Some students are 25 years old or older. September 11, / 10
11 Problems for Sept. 11, Let A = {1,2,3,4,5}. Determine the truth value of each of the following statements: (a) ( x A)(x + 3 = 10) (b) ( x A)(x + 3 < 10) (c) ( x A)(x + 3 < 5) (d) ( x A)(x + 3 7) 2 Determine the truth value of each of the following statements where U = {1,2,3} is the universal set: (a) x y. x 2 < y + 1 (b) x y. x 2 + y 2 < 12 (c) x y. x 2 + y 2 < 12 3 Negate each of the following statements: (a) x y. p(x, y) (b) x y. p(x, y) (c) y x z. p(x, y, z) 4 Negate each of the following statements: (a) All students live in the dormitories. (b) All mathematics majors are males. (c) Some students are 25 years old or older. September 11, / 10
12 Problems for Sept. 11, Let A = {1,2,3,4,5}. Determine the truth value of each of the following statements: (a) ( x A)(x + 3 = 10) (b) ( x A)(x + 3 < 10) (c) ( x A)(x + 3 < 5) (d) ( x A)(x + 3 7) 2 Determine the truth value of each of the following statements where U = {1,2,3} is the universal set: (a) x y. x 2 < y + 1 (b) x y. x 2 + y 2 < 12 (c) x y. x 2 + y 2 < 12 3 Negate each of the following statements: (a) x y. p(x, y) (b) x y. p(x, y) (c) y x z. p(x, y, z) 4 Negate each of the following statements: (a) All students live in the dormitories. (b) All mathematics majors are males. (c) Some students are 25 years old or older. September 11, / 10
13 Problems for Sept. 11, Let A = {1,2,3,4,5}. Determine the truth value of each of the following statements: (a) ( x A)(x + 3 = 10) (b) ( x A)(x + 3 < 10) (c) ( x A)(x + 3 < 5) (d) ( x A)(x + 3 7) 2 Determine the truth value of each of the following statements where U = {1,2,3} is the universal set: (a) x y. x 2 < y + 1 (b) x y. x 2 + y 2 < 12 (c) x y. x 2 + y 2 < 12 3 Negate each of the following statements: (a) x y. p(x, y) (b) x y. p(x, y) (c) y x z. p(x, y, z) 4 Negate each of the following statements: (a) All students live in the dormitories. (b) All mathematics majors are males. (c) Some students are 25 years old or older. September 11, / 10
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