Formal Logic - Lesson 6 - Switching Circuits

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FORMAL LOGIC
Discrete Structures I
FOR-IAN V. SANDOVAL

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LEARNING OBJECTIVES
❑ Apply mathematical logic to switching circuits

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SWITCHING CIRCUITS
❑ collection of wires and switches connecting two terminals, X
and Y
❑ A switch may be either open (O or 1) or closed (C or 0)
❑ An open switch will not permit the current to flow while a
closed switch will permit current to flow
X Y
Single Wire
X p Y
Open Switch (O) , p
X p Y
Closed Switch (O) , p
X p q Y
Series Switch: p ^ q Parallel Switch: p v q
X
p
q
Y

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SWITCHING CIRCUITS
❑ Two switches are complementary if one switch is open and
the other is closed, and vice versa
❑ Two switches are equivalent if they have the same electrical
properties concerning the flow and non-flow of current

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TRUTH TABLE FOR SWITCHING CIRCUITS
p q p ^ q p v q p q Series
Circuit
Parallel
Circuit
T T T T C C C C
T F F T or C O O C
F T F T O C O C
F F F F O O O O

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SWITCHING CIRCUITS
❑ Construct the switching circuits equivalent in each of the
following compound statements by applying the laws of
logical equivalence.
1. [(p ^ r) v (q ^ r)] v ~q

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SWITCHING CIRCUITS
1. [(p ^ r) v (q ^ r)] v ~q
[(p ^ r) v (q ^ r)] v ~q ≡
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v q) v ~q] ^ (r v ~q)
Distributive Law
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v (q v ~q)] ^ (r v ~q)
Associative Law
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v q) ^ r] v ~q
Distributive Law
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v q) ^ r] v ~q
Distributive Law
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v q) v ~q] ^ (r v ~q)
Distributive Law
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v (q v ~q)] ^ (r v ~q)
Associative Law
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v T)] ^ (r v ~q)
Inverse Law
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v T)] ^ (r v ~q)
Inverse Law
[(p ^ r) v (q ^ r)] v ~q ≡ T ^ (r v ~q)
Universal Bound Law
[(p ^ r) v (q ^ r)] v ~q ≡ r v ~q
Identity Law
[(p ^ r) v (q ^ r)] v ~q ≡ r v ~q
Identity Law

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SWITCHING CIRCUITS
1. [(p ^ r) v (q ^ r)] v ~q
[(p ^ r) v (q ^ r)] v ~q ≡ ~q v r
Commutative Law

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SWITCHING CIRCUITS
[(p ^ r) v (q ^ r)] v ~q
X
p r
Y

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SWITCHING CIRCUITS
[(p ^ r) v (q ^ r)] v ~q
X
p r
Yq r

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SWITCHING CIRCUITS
[(p ^ r) v (q ^ r)] v ~q
X
p r
q r
~q
Y

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SWITCHING CIRCUITS
~q v r
X
~q
Y
Simplified compound statement of [(p ^ r) v (q ^ r)] v ~q

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SWITCHING CIRCUITS
~q v r
X
~q
Y
r
Simplified compound statement of [(p ^ r) v (q ^ r)] v ~q

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Group Enrichment Exercises
❑ Construct the switching circuits equivalent in each of the
following compound statements by applying the laws of
logical equivalence.
1. [p v (~p v q) v (p v ~q)] ^ ~q
2. ~(p → q) ^ (p ↔ q)

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1. [p v (~p v q) v (p v ~q)] ^ ~q
[p v (~p v q) v (p v ~q)] ^ ~q ≡[p v (~p v q) v (p v ~q)] ^ ~q ≡[p v (~p v q) v (p v ~q)] ^ ~q ≡ [(p v ~p) v q v (p v ~q)] ^ ~q
Associative Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [(p v ~p) v q v (p v ~q)] ^ ~q
Associative Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [T v q v (p v ~q)] ^ ~q
Inverse Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [T v q v (p v ~q)] ^ ~q
Inverse Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [(T v q) v (p v ~q)] ^ ~q
Associative Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [(q v T) v (p v ~q)] ^ ~q
Commutative Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [T v (p v ~q)] ^ ~q
Universal Bound Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [T v (p v ~q)] ^ ~q
Universal Bound Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [(p v ~q) v T] ^ ~q
Commutative Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ T ^ ~q
Universal Bound Law

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GROUP ENRICHMENT EXERCISES
1. [p v (~p v q) v (p v ~q)] ^ ~q
[p v (~p v q) v (p v ~q)] ^ ~q ≡ ~q
Identity Law

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X
[p v (~p v q) v (p v ~q)] ^ ~q

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X
p
[p v (~p v q) v (p v ~q)] ^ ~q

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X
p
[p v (~p v q) v (p v ~q)] ^ ~q
~p
q

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X
p
[p v (~p v q) v (p v ~q)] ^ ~q
~p
q
p
~q

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GROUP ENRICHMENT EXERCISES
X
p
Y
[p v (~p v q) v (p v ~q)] ^ ~q
~p
q
p
~q
~q

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~q
X ~q Y
Simplified compound statement of [p v (~p v q) v (p v ~q)] ^ ~q

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2. ~(p → q) ^ (p ↔ q)
~(p → q) ^ (p ↔ q) ≡~(p → q) ^ (p ↔ q) ≡~(p → q) ^ (p ↔ q) ≡ ~(~p v q) ^ (p ↔ q)
Implication Law
~(p → q) ^ (p ↔ q) ≡ ~(~p v q) ^ (p ↔ q)
Implication Law
~(p → q) ^ (p ↔ q) ≡ [~(~p) ^ ~q)] ^ (p ↔ q)
De Morgan’s Law
~(p → q) ^ (p ↔ q) ≡ [~(~p) ^ ~q)] ^ (p ↔ q)
De Morgan’s Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (p ↔ q)
Double Negation Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (p ↔ q)
Double Negation Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ [(~p v q) ^ (p v ~q)]
Equivalence Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (p v ~q) ^ (~p v q)
Commutative Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (p v ~q) ^ (~p v q)
Commutative Law
~(p → q) ^ (p ↔ q) ≡ [(p ^ ~q) ^ (p v ~q)] ^ (~p v q)
Commutative Law
~(p → q) ^ (p ↔ q) ≡ {(p ^ [~q ^ (p v ~q)]} ^ (~p v q)
Associative Law
~(p → q) ^ (p ↔ q) ≡ {(p ^ [~q ^ (p v ~q)]} ^ (~p v q)
Associative Law

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2. ~(p → q) ^ (p ↔ q)
~(p → q) ^ (p ↔ q) ≡ {(p ^ [~q ^ (~q v p)]} ^ (~p v q)
Commutative Law
~(p → q) ^ (p ↔ q) ≡ {(p ^ [~q ^ (~q v p)]} ^ (~p v q)
Commutative Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (~p v q)
Absorption Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (~p v q)
Absorption Law
~(p → q) ^ (p ↔ q) ≡ p ^ [~q ^ (~p v q)]
Associative Law
~(p → q) ^ (p ↔ q) ≡ p ^ [(~q ^ ~p) v (~q ^ q)]
Distributive Law
~(p → q) ^ (p ↔ q) ≡ p ^ [(~q ^ ~p) v (~q ^ q)]
Distributive Law
~(p → q) ^ (p ↔ q) ≡ p ^ [(~q ^ ~p) v (q ^ ~q)]
Commutative Law
~(p → q) ^ (p ↔ q) ≡ p ^ [(~q ^ ~p) v F]
Inverse Law
~(p → q) ^ (p ↔ q) ≡ p ^ (~q ^ ~p)
Identity Law
~(p → q) ^ (p ↔ q) ≡ p ^ (~q ^ ~p)
Identity Law
~(p → q) ^ (p ↔ q) ≡ p ^ [(~q ^ ~p) v F]
Inverse Law

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2. ~(p → q) ^ (p ↔ q)
~(p → q) ^ (p ↔ q) ≡ p ^ (~p ^ ~q )
Commutative Law
~(p → q) ^ (p ↔ q) ≡ p ^ (~p ^ ~q )
Commutative Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~p) ^ ~q
Associate Law
~(p → q) ^ (p ↔ q) ≡ F ^ ~q
Inverse Law
~(p → q) ^ (p ↔ q) ≡ ~q ^ F
Commutative Law
~(p → q) ^ (p ↔ q) ≡ F
Universal Bound Law

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(p ^ ~q) ^ (p v ~q) ^ (~p v q)
X
Applying the Laws of Equivalence to
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (~p v q) ^ (p v ~q)
p ~q

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(p ^ ~q) ^ (p v ~q) ^ (~p v q)
X
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (~p v q) ^ (p v ~q)
p ~q
p
~q

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(p ^ ~q) ^ (p v ~q) ^ (~p v q)
X
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (~p v q) ^ (p v ~q)
p ~q
p
~q
~p
q

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(p ^ ~q) ^ (p v ~q) ^ (~p v q)
X
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (~p v q) ^ (p v ~q)
p ~q
p
~q
~p
q
Y

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SWITCHING CIRCUITS
• A number of different patterns of open and closed switches
that will allow the current to flow from X and Y.
• In example, using the group enrichment exercises no. 1
• The several patterns are illustrated in the following slides.

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SWITCHING CIRCUITS
• Case 1: When p = T or closed and q = T or closed
X
p
Y
[p v (~p v q) v (p v ~q)] ^ ~q
~p
q
p
~q
~q

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SWITCHING CIRCUITS
• Case 2: When p = T or closed and q = F or open
X
p
Y
[p v (~p v q) v (p v ~q)] ^ ~q
~p
q
p
~q
~q

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SWITCHING CIRCUITS
• Case 3: When p = F or open and q = T or closed
X
p
Y
[p v (~p v q) v (p v ~q)] ^ ~q
~p
q
p
~q
~q

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SWITCHING CIRCUITS
• Case 4: When p = F or open and q = F or open
X
p
Y
[p v (~p v q) v (p v ~q)] ^ ~q
~p
q
p
~q
~q

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Group Reinforcement Activity
Construct the switching circuits in each of the following
equivalent compound statements..
1. (p v q) ^ r ^ q
2. [(p v q) ^ r] ^ q
3. (p v r v ~s) ^ (p v q v r) ^ (p v ~q v s)

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1. (p v q) ^ r ^ q
X
p
q
(p v q) ^ r ^ q

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1. (p v q) ^ r ^ q
X
p
q
(p v q) ^ r ^ q
r

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1. (p v q) ^ r ^ q
X
p
q
(p v q) ^ r ^ q
r q Y

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X
p
q
(p v q) ^ r ^ q
r q Y
Case 1: When p = T or closed, q = T or closed and
r = T or closed

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X
p
q
(p v q) ^ r ^ q
r q Y
Case 2: When p = T or closed, q = T or closed and
r = F or open

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X
p
q
(p v q) ^ r ^ q
r q Y
Case 3: When p = T or closed, q = F or open and
r = F or open

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X
p
q
(p v q) ^ r ^ q
r q Y
Case 4: When p = F or open, q = F or open and
r = F or open

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X
p
q
(p v q) ^ r ^ q
r q Y
Case 5: When p = F or open, q = T or closed and
r = F or open

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X
p
q
(p v q) ^ r ^ q
r q Y
Case 6: When p = F or open, q = T or closed and
r = T or closed

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X
p
q
(p v q) ^ r ^ q
r q Y
Case 7: When p = F or open, q = F or open and
r = T or closed

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2. [(p v q) ^ r] ^ q
X
p
q
[(p v q) ^ r] ^ q

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2. [(p v q) ^ r] ^ q
X
p
q
[(p v q) ^ r] ^ q
r

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2. [(p v q) ^ r] ^ q
X
p
q
[(p v q) ^ r] ^ q
r q Y

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Show the different patterns that will allow the current to flow
from X to Y.
X
p
q
[(p v q) ^ r] ^ q
r q Y

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X
p
v
~s
(p v r v ~s) ^ (p v q v r) ^ (p v ~q v s)

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X
p
v
~s
p
q
r

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X
p
v
~s
p
q
r
p
~q
s
Y

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Show the different patterns that will allow the current to flow
from X to Y.
X
p
v
~s
p
q
r
p
~q
s
Y

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• Levin, O. (2019). Discrete Mathematics: An Open Introduction 3rd Edition. Colorado: School of Mathematics Science
University of Colorado.
• Aslam, A. (2016). Proposition in Discrete Mathematics retrieved from https://www.slideshare.net/AdilAslam4/chapter-1-
propositions-in-discrete-mathematics
• Operator Precedence retrieved from http://intrologic.stanford.edu/glossary/operator_precedence.html
REFERENCES

Formal Logic - Lesson 6 - Switching Circuits

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Formal Logic - Lesson 6 - Switching Circuits