Discreet Structure/Math Propositional/Predicate Logic HW help

1. If A is true, B is false and C is false what is the truth value of the following wff

(a) A ( B C )

(b) ( B A ) C

(c) A (B C )

(d) (A B ) C

(e) A (B C )

2. Given the following statements

A: Roses are red

B: Violets are blue

S: Sugar is sweet

Convert the following sentences to symbolic notation:

(a) Roses are red and violets are blue

(b) Roses are red and either violets are blue or sugar is not sweet

(c) If violets are blue, then roses are red and sugar is sweet

(d) Sugar is not sweet if and only if roses are not red and violets are blue

(e) Roses are red and, if sugar is not sweet, t hen either violets are not blue or sugar is sweet.

3. Construct the truth tables for the given wff. Note any tautologies or contradictions

(a) A (B A)

(b) A B A

(c) [ (A B) C ] A C

(d) (A B) [ (A C) (B C) ]

(e) A B A B

4. Use propositional logic to prove the following arguments are valid

(a) (P (Q R)) ( R S ) ( S T ) (T P)

(b) (A B) (B A ) (C B )

(c) (A B) B ( A C) C

(d) (A B ) (B C) (A C)

(e) (A B) [ A ( B C)] (A C )

5. Write the argument using propositional wffs (using the statement letters shown) . Then prove the argument is valid using propositional logic.

(a) If chicken is on the menu, then dont order fish, but you should have either fish or salad. So if chicken is on the menu have salad. (C, F, S)

(b) If a Democrat is elected then taxes will go up. Either a Democrat will be elected or the bill will pass. Therefore if taxes do not go up then the bill will pass. (D, T, B)

(c) It is not the case that if electric rates go up, then usage will go down, nor is it true that either new power plants will be built or bills will not be late. Therefore usage will not go down and bills will be late. (R, U, P, B)

(d) If birds are flying south and the leaves are turning, then it must be fall. Fall brings cold weather. The leaves are turning but the weather is not cold. Therefore, the birds are not flying south. (B, L, F, C)

(e) If Jane is more popular, then she will be elected. If Jane is more popular, the Craig will resign. Therefore if Jane is more popular, she will be elected and Craig will resign. (J, E, C)

6. Give the truth value of each of the following wffs in the interpretation where the domain consists of people, M(x,y ) is x is the mother of y, F( x) is x is female, M(x) is x is male.

(a) ( x) ( y) ( M(y, x))

(b) ( x) ( y) (M (x, y))

(c) ( x) ( y) (M(x, y) M(y))

(d) ( x) ( y) (M(x, y) M(y))

(e) ( x) ( y) (M(x, y) F(y))

7.Using the predicate symbols shown and appropria te quantifiers, write each English language statement as a predicate wff. (The domain is the whole world).

D(x): x is a day

S(x): x is sunny

R(x): x is rainy

m: Monday

t: Tuesday The sentences are:

(a) Every day that is sunny is not rainy

(b) Some days are sunny and rainy

(c) Monday was sunny; therefore every day will be sunny

(d) It rained on both Monday and Tuesday

(e) If some day is rainy then every day will be sunny