Use a truth table to verify the distributive law p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r).

Use a truth table to verify the first De Morgan law ¬(p ∧ q) ≡ ¬p ∨ ¬q.

Use truth tables to verify these equivalences.

a) p ∧ T ≡ p

b) p v F ≡ p

c) p ∧ F ≡ F

d) p v T ≡ T

e) p v p ≡ p

f) p ∧ p ≡ p

Show that ¬(¬p) and p are logically equivalent.

Use truth tables to check these laws:

(a) The second DeMorgan's law. (The first was checked in the text.)

(b) The distributive laws.

Construct a truth table for each of these compound propositions. a) (p ∨ q) → (p ⊕ q) b) (p ⊕ q) → (p ∧ q) c) (p ∨ q) ⊕ (p ∧ q) d) (p ↔ q) ⊕ (¬p ↔ q) e) (p ↔ q) ⊕ (¬p ↔ ¬r) f) (p ⊕ q) → (p ⊕ ¬q)

Show that (p ∧ q) → r and (p → r) ∧ (q → r) are not logically equivalent.

Construct a truth table for each of these compound propositions. a) p → ¬p b) p ↔ ¬p c) p ⊕ (p ∨ q) d) (p ∧ q) → (p ∨ q) e) (q → ¬p) ↔ (p ↔ q) f) (p ↔ q) ⊕ (p ↔ ¬q)

Verify the first Absorption Law by means of a truth table.

(a) Make up three simple statements and label them p, q and r. Then write compound statements to represent (pVq)/\r and pV(q/\r). (b) Do you think that the statements for (p∨q)/r and p∨(q/\r) mean the same thing? Explain.

Show that (p → q) ∧ (q → r) → (p → r) is a tautology.

Construct a truth table for each compound statement. p and q

Express these system specifications using the propositions p “The message is scanned for viruses” and q “The message was sent from an unknown system” together with logical connectives (including negations). a) “The message is scanned for viruses whenever the message was sent from an unknown system.” b) “The message was sent from an unknown system but it was not scanned for viruses.” c) “It is necessary to scan the message for viruses whenever it was sent from an unknown system.” d) “When a message is not sent from an unknown system it is not scanned for viruses.”

Prove the first distributive law by showing that if A, B, and C are sets, then A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).