Logic and Proposition Simplification: A Study of Truth Statements, Symbolic Representation, and Proof Techniques
This assignment focuses on logic, symbolic representation, and proof techniques in mathematics.
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Logic and Proposition Simplification: A Study of Truth Statements,
Symbolic Representation, and Proof Techniques
1.
Which of the following are statements?
1. She is a mathematics major.
2. 128=26
3. All that glitters is not gold.
4. Sleep tight and don’t let the bedbugs bite.
Soln. A sentence that can be judged to be true or false is called a statement.
1. She is a mathematics major is a statement as it can be either true or false.
2. 128=26 is a statement as it is false.
3. All that glitters is not gold is a statement as it can be either true or false.
4. Sleep tight and don’t let the bedbugs bite is not a statement.
2.
Let A, B, and C be the following statements:
A: John is healthy
B: John is wealthy
C: John is wise
Use A, B, and C as defined above to translate the following statements into symbolic form.
1. John is not wealthy but he is healthy and wise.
(B′) ∧ (A ∧ C)
2. John is neither healthy, wealthy, nor wise.
A′ ∧ B′ ∧ C′
3. John is wealthy, but he is not both healthy and wise.
B ∧ (A ∧ C)′
3.
Simplify the following proposition to 2 logic operations using the laws of the algebra of
propositions. Write each step on a separate line with the algebra law you used as a
justification. Missing steps will be penalized.
(P′∧Q′) ∨ (P′∧Q) ∨ (P∧Q′)
Soln. By identity P= (P∨P) and associativity we have,
(P′∧Q′) ∨ (P′∧Q) ∨ (P′∧Q′) ∨ (P∧Q′)
By distributivity we have,
(P′∧(Q′∨Q)) ∨ ((P′∨P)∧Q′)
Reduction by the law of excluded middle P′∨P = 1 we get,
(P′∧1) ∨ (1∧Q′)
By the neutral element definition P′∧1 = P we get,
Symbolic Representation, and Proof Techniques
1.
Which of the following are statements?
1. She is a mathematics major.
2. 128=26
3. All that glitters is not gold.
4. Sleep tight and don’t let the bedbugs bite.
Soln. A sentence that can be judged to be true or false is called a statement.
1. She is a mathematics major is a statement as it can be either true or false.
2. 128=26 is a statement as it is false.
3. All that glitters is not gold is a statement as it can be either true or false.
4. Sleep tight and don’t let the bedbugs bite is not a statement.
2.
Let A, B, and C be the following statements:
A: John is healthy
B: John is wealthy
C: John is wise
Use A, B, and C as defined above to translate the following statements into symbolic form.
1. John is not wealthy but he is healthy and wise.
(B′) ∧ (A ∧ C)
2. John is neither healthy, wealthy, nor wise.
A′ ∧ B′ ∧ C′
3. John is wealthy, but he is not both healthy and wise.
B ∧ (A ∧ C)′
3.
Simplify the following proposition to 2 logic operations using the laws of the algebra of
propositions. Write each step on a separate line with the algebra law you used as a
justification. Missing steps will be penalized.
(P′∧Q′) ∨ (P′∧Q) ∨ (P∧Q′)
Soln. By identity P= (P∨P) and associativity we have,
(P′∧Q′) ∨ (P′∧Q) ∨ (P′∧Q′) ∨ (P∧Q′)
By distributivity we have,
(P′∧(Q′∨Q)) ∨ ((P′∨P)∧Q′)
Reduction by the law of excluded middle P′∨P = 1 we get,
(P′∧1) ∨ (1∧Q′)
By the neutral element definition P′∧1 = P we get,
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Subject
Mathematics