Homework answers / question archive / CSS 220 Module 5 In-Class Problems PROPOSITIONAL LOGIC: Computer Circuits p,q,r – input nodes s,c – output nodes    - XOR gate (the output of an XOR gate is on if and only if its inputs disagree with each other

CSS 220 Module 5 In-Class Problems PROPOSITIONAL LOGIC: Computer Circuits p,q,r – input nodes s,c – output nodes    - XOR gate (the output of an XOR gate is on if and only if its inputs disagree with each other

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CSS 220 Module 5 In-Class Problems

PROPOSITIONAL LOGIC: Computer Circuits

p,q,r – input nodes

s,c – output nodes

 

 - XOR gate (the output of an XOR gate is on if and only if its inputs disagree with each other.) ¬ (p ∨ q)

 

 - AND gate (the output of an AND gate is on if and only if both of its inputs are on)

p q

 - OR gates (the value of an OR node is on if and only if at least one of its inputs is on)

p ∨ q

 

 

 

 

1. Identify which of the following are propositions: (circle one)

a. p: Today is Friday PROPOSITION NOT A PROPOSITION

b. p: 3 + 5 = 8 PROPOSITION NOT A PROPOSITION

c. p: Take the quiz PROPOSITION NOT A PROPOSITION

d. p: 7 < 11 PROPOSITION NOT A PROPOSITION

e. p: Put your hat on PROPOSITION NOT A PROPOSITION

f. p: a triangle has 4 sides PROPOSITION NOT A PROPOSITION

2. What is the difference between the truth value and the truth table (Semantics).

 

3. NEGATION (NOT): The negation of a proposition can be formed by inserting the word _______ as appropriate. The notation for the negation of p is p.

 

Example:

State the negation of the following propositions:

a. p: Today is Saturday. p: _______________________________

b. p: All mammals respire p: _______________________________

c. p: The glass is full p: _______________________________

 

 

The truth table for a negation is:

 

q	¬q

 

 

4. p in logic corresponds with _________ in set

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5. CONJUNCTION (AND): A conjunction is formed when two propositions are connected by the word _________.

 

Example:

Let p: London is the capital of England.

q: Houston is the capital of the United States.

State p q: ___________________________________________________

 

The truth table for a conjunction is:

 

p	q	p q

 

 

 

6. Tautology vs Contradiction Examples:

 

 

 

 

 

7. DISJUNCTION (OR): A disjunction is formed when propositions are joined by the word “or.” Example:

Let p: London is the capital of England.

q: Houston is the capital of the United States.

State p q: ________________________________________________

 

 

The truth table for a conjunction is:

 

p	q	p V q

 

 

 

 

8. Populate this truth table for ¬p ∨ ¬q

 

p	q	¬p	¬q	¬p ∨ ¬q

 

 

 

9. If you have n propositions, how many lines will you have in your truth table?

 

 

10. Create a truth table for (p q) r

p	q	r	(p q)	(p q)	(p q) r

 

 

11. IMPLICATION (If-Then): When a proposition p being true implies that another proposition q must also be true then we say that p implies q. p ⇒ q

 

Example:

a. If you score 90% or above in this class, then you will get an A.

b. My thumb will hurt if I hit it with a hammer.

c. x = 2 implies x + 1 = 3.

d. If Jimmy loses a tooth, then Jimmy finds a dollar.

 

 

p (antecedent)	q (consequence)	p implies q
T	T	T
T	F	F
F	T	T
F	F	T

 

 

 

 

12. EQUIVALENCE (If-And-Only-If): Two propositions p and q are called equivalent statements if each implies the other (p if and only if q): p ≡ q, p↔q

p: we will go to the amusement park,

q: we will go to the zoo

 

p	q	p ↔ q
T	T
T	F
F	T
F	F

 

 

 

 

13. Prove that:

p ⇒ q ≡ ¬p ∨ q

 

 

p	q	p ⇒ q	¬ p	¬p ∨ q

 

 

14. Distributive Law p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (q ∧ r)

p	q

 

 

 

 

 

 

15. De Morgan’s Law: ¬(p ∨ q) ≡ ¬p ¬q

 

p	q

 

 

16. Simplify: (p ∧ q) ∨ ( ¬q ∧ p)f