Conditional Statements
Lecture 3
Section 2.2
Robb T. Koether
Hampden-Sydney College
Fri, Jan 17, 2014
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
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1
Conditional Statements
2
The Contrapositive
3
Other Operators
The Biconditional
The Exclusive-Or
The NAND Operator
The NOR Operator
4
Assignment
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
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Outline
1
Conditional Statements
2
The Contrapositive
3
Other Operators
The Biconditional
The Exclusive-Or
The NAND Operator
The NOR Operator
4
Assignment
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
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Conditional Statements
A conditional statement is a statement of the form
p → q.
p is the hypothesis.
q is the conclusion.
Read p → q as “p implies q” or “if p, then q.”
The idea is that the truth of p implies the truth of q (but nothing
more).
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Conditional Statements
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Example: Conditional Statements
Which of the following scenarios are consistent with the statement
“If I win the lottery, then I will buy a Mercedes Benz.”
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
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Example: Conditional Statements
Which of the following scenarios are consistent with the statement
“If I win the lottery, then I will buy a Mercedes Benz.”
Scenario 1: I win the lottery and I buy a Mercedes Benz.
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
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Example: Conditional Statements
Which of the following scenarios are consistent with the statement
“If I win the lottery, then I will buy a Mercedes Benz.”
Scenario 1: I win the lottery and I buy a Mercedes Benz.
Scenario 2: I win the lottery and I do not buy a Mercedes Benz.
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
5 / 26
Example: Conditional Statements
Which of the following scenarios are consistent with the statement
“If I win the lottery, then I will buy a Mercedes Benz.”
Scenario 1: I win the lottery and I buy a Mercedes Benz.
Scenario 2: I win the lottery and I do not buy a Mercedes Benz.
Scenario 3: I do not win the lottery and I do not buy a Mercedes
Benz.
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
5 / 26
Example: Conditional Statements
Which of the following scenarios are consistent with the statement
“If I win the lottery, then I will buy a Mercedes Benz.”
Scenario 1: I win the lottery and I buy a Mercedes Benz.
Scenario 2: I win the lottery and I do not buy a Mercedes Benz.
Scenario 3: I do not win the lottery and I do not buy a Mercedes
Benz.
Scenario 4: I do not win the lottery and I buy a Mercedes Benz.
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
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Truth Table for the Conditional
p
T
T
F
F
q
T
F
T
F
p→q
T
F
T
T
p → q is true if p is false or q is true.
p → q is false if p is true and q is false.
Thus, p → q is logically equivalent to ∼ p ∨ q.
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
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Outline
1
Conditional Statements
2
The Contrapositive
3
Other Operators
The Biconditional
The Exclusive-Or
The NAND Operator
The NOR Operator
4
Assignment
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
7 / 26
The Contrapositive
The contrapositive of p → q is ∼ q →∼ p.
The statements p → q and ∼ q →∼ p are logically equivalent.
(Prove it!)
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Conditional Statements
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The Converse and the Inverse
p→q
q→p
∼p → ∼q
∼q → ∼p
Four conditionals.
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Conditional Statements
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The Converse and the Inverse
p→q
∼p → ∼q
Converses
Converses
q→p
∼q → ∼p
The converse of p → q is q → p
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Conditional Statements
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The Converse and the Inverse
p→q
q→p
Inverses
Inverses
∼p → ∼q
∼q → ∼p
The inverse of p → q is ∼ p →∼ q
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Conditional Statements
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The Converse and the Inverse
p→q
q→p
Contrapositives
∼p → ∼q
∼q → ∼p
The contrapositive of p → q is ∼ q →∼ p
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Conditional Statements
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Outline
1
Conditional Statements
2
The Contrapositive
3
Other Operators
The Biconditional
The Exclusive-Or
The NAND Operator
The NOR Operator
4
Assignment
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
10 / 26
Outline
1
Conditional Statements
2
The Contrapositive
3
Other Operators
The Biconditional
The Exclusive-Or
The NAND Operator
The NOR Operator
4
Assignment
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
11 / 26
The Biconditional
The biconditional of p and q is denoted p ↔ q.
Read p ↔ q as “p if and only if q.”
p ↔ q is logically equivalent to
(p → q) ∧ (q → p).
It is also logically equivalent to
(∼ p ∨ q) ∧ (∼ q ∨ p)
and
(p ∧ q) ∨ (∼ p ∧ ∼ q).
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
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The Biconditional
p
T
T
F
F
Robb T. Koether (Hampden-Sydney College)
q
T
F
T
F
p↔q
T
F
F
T
Conditional Statements
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Outline
1
Conditional Statements
2
The Contrapositive
3
Other Operators
The Biconditional
The Exclusive-Or
The NAND Operator
The NOR Operator
4
Assignment
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
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Exclusive-Or
The exclusive-or of p and q is denoted p ⊕ q.
p ⊕ q means “one or the other, but not both.”
p ⊕ q is logically equivalent to
(p ∨ q) ∧ ∼ (p ∧ q)
and
(p ∧ ∼ q) ∨ (∼ p ∧ q)
and
∼ (p ↔ q).
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
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Exclusive-Or
p
T
T
F
F
Robb T. Koether (Hampden-Sydney College)
q
T
F
T
F
p↔q
F
T
T
F
Conditional Statements
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Outline
1
Conditional Statements
2
The Contrapositive
3
Other Operators
The Biconditional
The Exclusive-Or
The NAND Operator
The NOR Operator
4
Assignment
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
17 / 26
The NAND Operator
p
T
T
F
F
q
T
F
T
F
p|q
F
T
T
T
The NAND of p and q is denoted p | q.
The operator | is also called the Scheffer stroke.
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
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The NAND Operator
The statement p | q means “not both p and q.”
p | q is logically equivalent to ∼ (p ∧ q).
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
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The NAND Operator
The three basic operators (and, or, not) may be defined in terms
of NAND.
∼ p ≡ p | p.
p ∧ q ≡ (p | q) | (p | q).
p ∨ q ≡ (p | p) | (q | q).
Prove it!
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
20 / 26
Outline
1
Conditional Statements
2
The Contrapositive
3
Other Operators
The Biconditional
The Exclusive-Or
The NAND Operator
The NOR Operator
4
Assignment
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
21 / 26
The NOR Operator
p
T
T
F
F
q
T
F
T
F
p↓q
F
F
F
T
The NOR of p and q is denoted p ↓ q.
The operator ↓ is also called the Pierce arrow.
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
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The NOR Operator
The statement p ↓ q means “neither p nor q.”
p ↓ q is logically equivalent to ∼ (p ∨ q).
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
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The NOR Operator
The three basic operators (and, or, not) may be defined in terms
of NOR.
∼ p ≡ p ↓ p.
p ∨ q ≡ (p ↓ q) ↓ (p ↓ q).
p ∧ q ≡ (p ↓ p) ↓ (q ↓ q).
Prove it!
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
24 / 26
Outline
1
Conditional Statements
2
The Contrapositive
3
Other Operators
The Biconditional
The Exclusive-Or
The NAND Operator
The NOR Operator
4
Assignment
Robb T. Koether (Hampden-Sydney College)
Conditional Statements
Fri, Jan 17, 2014
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Assignment
Assignment
Read Section 2.2, pages 39 - 48.
Exercises 2, 5, 6, 12, 13, 17, 18, 37, 41, 42, 46, page 49.
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