1. A Brief Biography of George Boole

2. George Boole, an English Mathematician, published a book in 1854 called “An investigation into the Laws of Thought”. This was his most important work.

3. His work showed the use of logic in mathematics and the implications it has on data. More specifically, logical calculus with only two values, “true” or “false”.
His findings paved the way to computers. Without Boolean logic, there would be no computers today.
Born: 2 November 1815
Died: 8 December (pneumonia)
Birthplace: Lincoln, England
Best known as: Developer of Boolean algebra

4. Show these two circuits have logically equivalent Boolean expressions.
Sec. 2.4 #28
Show these two circuits have logically equivalent Boolean expressions.

5. Solution Using Theorem 2.1.1 Logical Equivalences
Circuit “A” and Circuit “B” represented as a Boolean Expression:
(P ∧ Q) ∨ (P ∧ ∼Q) ∨ (∼P ∧ ∼Q) ≡ P ∨ ∼Q
(P ∧ Q) ∨ (P ∧ ∼Q) ∨ (∼P ∧ ∼Q) ≡ P ∨ ∼Q
P ∧ (Q ∨ ∼Q) ∨ (∼P ∧ ∼Q) ≡ P ∨ ∼Q Distribution
P ∧ t ∨ (∼P ∧ ∼Q) ≡ P ∨ ∼Q Negation
P ∨ (∼P ∧ ∼Q) ≡ P ∨ ∼Q Identity
(P ∨ ∼P) ∧ (P ∨ ∼Q) ≡ P ∨ ∼Q Distribution
t ∧ (P ∨ ∼Q) ≡ P ∨ ∼Q Negation
P ∨ ∼Q ≡ P ∨ ∼Q Identity

6. Question #32 Sec. 2.4
Find a circuit with at most three logic gates that is equivalent to this circuit.
( P ^ Q ^ R ) OR ( P ^ ~ Q ^ R ) OR ( P ^ ~ Q ^ ~ R )

7. Solution

8. Section 2.4 Problem # 33 Part a) Show that for the Sheffer Stroke |,
P ⋀ Q ≣ (P∣Q)|(P|Q)
P ⋀ Q ≣ P ⋁(P ⋀ Q) ⋀ Q ⋁(Q ⋀ P) Absorption Law
≣ P ⋁ P ⋀ Q ⋀ Q ⋁ P ⋀ Q Commutative Law
≣ (P ⋁ P) ⋀ (Q ⋀ Q) ⋁ (P ⋀ Q)
≣ (P ⋀ Q) ⋁ (P ⋀ Q) Idempotent Law
≣ ~ ( ~ ((P ⋀ Q) ⋁ (P ⋀ Q))) Double negative Law
≣ ~ (~(P ⋀ Q) ⋀ ~(P ⋀ Q)) De Morgan’s Law
≣ ~ ((P|Q) ⋀ (P|Q)) Definition of |
P ⋀ Q ≣ (P|Q)|(P|Q) Definition of |

9. (P|[(~Q|~Q)|(R|R)])|(P|[(~Q|~Q)|(R|R)])|
Part b) Use the results of Example and part (a)
above to write P ⋀ (~Q ⋁ R) using only Sheffer strokes.
P ⋀ (~Q ⋁ R)
P ⋀ ((~Q|~Q)|(R|R))
(P|[(~Q|~Q)|(R|R)])|(P|[(~Q|~Q)|(R|R)])|

10. Henry Maurice Sheffer
Henry M. Sheffer was born in the Ukraine in 1882, then migrated to the United States in Henry’s undergraduate work was done at Boston Latin School. He then studied at Harvard where he received his Ph.D. in Philosophy. After working at different universities Henry returned to Harvard; where he taught mathematical logic for 32 years.
Henry M. Sheffer is most well known for his work in Boolean algebra. In particular for creating the binary operation abbreviated as NAND and its dual called NOR.

11. Write the negation for the definition of
Sec 3.3 #43
Write the negation for the definition of
Definition:
Negation:
For all real numbers δ ≤ 0, there exists some real number ε ≤ 0 such that for some real number x, a – δ < x < a + δ and x ≠ a and L – ε ≥ f(x) ≥ L + ε.