1. Discrete Math for CS CMPSC 360 LECTURE 4 Last time and recitations:
Propositions with quantifiers
Proofs
Today:
More proofs
CMPSC
360
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2. Proof types Direct proof.
Proof by contradiction (also, proof by contraposition).
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3. Some useful definitions
An integer is even if 𝑛=2𝑘 for some integer 𝑘.
An integer is odd if 𝑛=2𝑘+1 for some integer 𝑘.
Two integers have the same parity if they are both even or both odd. Otherwise, they have opposite parity.
Integer 𝑎 divides integer 𝑏, written 𝒂|𝒃, if 𝑏=𝑘𝑎 for some integer 𝑘. Then 𝑎 is a divisor of 𝑏, and 𝑏 is a multiple of 𝑎.
A natural number 𝑛 is prime if it has exactly two positive divisors, 1 and 𝑛.
A natural number 𝑛 is composite if it factors into 𝑛=𝑎𝑏 for 𝑎>1,𝑏>1.
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4. I-clicker problem (frequency: BC)
Theorem. If 𝑛 is odd then 𝑛 2 is odd.
Which of the following arguments correctly prove the theorem?
Suppose 𝑛 is not odd. Then 𝑛=2𝑘 for some 𝑘. Then 𝑛 2 = 2𝑘 2 =2⋅ 2 𝑘 2 , which is not odd.
Suppose 𝑛 is odd. Then 𝑛=2𝑘+1 for some 𝑘. Then 𝑛 2 = 2𝑘+1 2 =4 𝑘 2 +4𝑘+1=2⋅ 2 𝑘 2 +2𝑘 +1, which is odd.
Both A and B.
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5. I-clicker problem (frequency: BC)
Theorem. A product of several numbers of the form (4𝑛+1) also has that form.
Indicate the first incorrect step in the following ``proof’’:
Suppose the numbers are 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑘 It is sufficient to prove the statement for a product of two numbers and apply it first to 𝑥 1 𝑥 2 , then to (𝑥 1 𝑥 2 ) 𝑥 3 , etc.
Consider two numbers of the specified form: 𝑥 1 =4𝑛+1 and 𝑥 2 =4𝑛+1.
Their product is 4𝑛+1 2 =16 𝑛 2 +8𝑛 =4 8 𝑛 2 +2𝑛 +1.
We showed that the product is of the right form.
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