Direct Proofs Let P and Q be given statements, and suppose we want to prove: “If P , then Q” or P ⇒ Q.

Direct Proofs Let P and Q be given statements, and suppose we want to prove: “If P , then Q” or P ⇒ Q.

1. Explore it.

Direct Proofs Let P and Q be given statements, and suppose we want to prove: “If P , then Q” or P ⇒ Q.

1. Explore it. 2. A direct proof begins by assuming the hypothesis P is true, followed by a logical sequence of steps, and ends by showing the conclusion Q is true.

Direct Proofs Let P and Q be given statements, and suppose we want to prove: “If P , then Q” or P ⇒ Q.

1. Explore it. 2. A direct proof begins by assuming the hypothesis P is true, followed by a logical sequence of steps, and ends by showing the conclusion Q is true. 3. We justify each step using definitions or previous knowledge.

Definition: Let n be an integer. Then: • n is even if ???

Definition: Let n be an integer. Then: • n is even if there is an integer k so that n = 2k,

Definition: Let n be an integer. Then: • n is even if there is an integer k so that n = 2k, and • n is odd if ???

Definition: Let n be an integer. Then: • n is even if there is an integer k so that n = 2k, and • n is odd if there is an integer k so that n = 2k + 1.

Definition: Let n be an integer. Then: • n is even if there is an integer k so that n = 2k, and • n is odd if there is an integer k so that n = 2k + 1. Example: Determine if the following statements are TRUE or FALSE.

Definition: Let n be an integer. Then: • n is even if there is an integer k so that n = 2k, and • n is odd if there is an integer k so that n = 2k + 1. Example: Determine if the following statements are TRUE or FALSE. 1. If x is an odd integer and y is an integer, then xy is odd.

Definition: Let n be an integer. Then: • n is even if there is an integer k so that n = 2k, and • n is odd if there is an integer k so that n = 2k + 1. Example: Determine if the following statements are TRUE or FALSE. 1. If x is an odd integer and y is an integer, then xy is odd. This is a FALSE statement since 3 is odd, 2 is an integer, and 3 · 2 = 6 is even, not odd.

Definition: Let n be an integer. Then: • n is even if there is an integer k so that n = 2k, and • n is odd if there is an integer k so that n = 2k + 1. Example: Determine if the following statements are TRUE or FALSE. 1. If x is an odd integer and y is an integer, then xy is odd. This is a FALSE statement since 3 is odd, 2 is an integer, and 3 · 2 = 6 is even, not odd. 2. If n is an odd integer, then 3n + 7 is even.

Definition: Let n be an integer. Then: • n is even if there is an integer k so that n = 2k, and • n is odd if there is an integer k so that n = 2k + 1. Example: Determine if the following statements are TRUE or FALSE. 1. If x is an odd integer and y is an integer, then xy is odd. This is a FALSE statement since 3 is odd, 2 is an integer, and 3 · 2 = 6 is even, not odd. 2. If n is an odd integer, then 3n + 7 is even. This could be TRUE. Let’s try to make a “Know-Show” table.

If n is odd, then 3n + 7 is even.

If n is odd, then 3n + 7 is even. Proof:

If n is odd, then 3n + 7 is even. Proof: We suppose ...

If n is odd, then 3n + 7 is even. Proof: We suppose that n is an odd integer and want to prove that 3n + 7 is even.

If n is odd, then 3n + 7 is even. Proof: We suppose that n is an odd integer and want to prove that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1.

If n is odd, then 3n + 7 is even. Proof: We suppose that n is an odd integer and want to prove that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7

If n is odd, then 3n + 7 is even. Proof: We suppose that n is an odd integer and want to prove that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7 3n + 7 = 6k + 10

If n is odd, then 3n + 7 is even. Proof: We suppose that n is an odd integer and want to prove that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7 3n + 7 = 6k + 10 3n + 7 = 2(3k + 5).

If n is odd, then 3n + 7 is even. Proof: We suppose that n is an odd integer and want to prove that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7 3n + 7 = 6k + 10 3n + 7 = 2(3k + 5). Since k is an integer, 3k + 5 is also an integer because integers are closed under addition and multiplication.

If n is odd, then 3n + 7 is even. Proof: We suppose that n is an odd integer and want to prove that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7 3n + 7 = 6k + 10 3n + 7 = 2(3k + 5). Since k is an integer, 3k + 5 is also an integer because integers are closed under addition and multiplication. If we let q be the integer 3k + 5, then by substitution we have shown 3n + 7 = 2q.

If n is odd, then 3n + 7 is even. Proof: We suppose that n is an odd integer and want to prove that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7 3n + 7 = 6k + 10 3n + 7 = 2(3k + 5). Since k is an integer, 3k + 5 is also an integer because integers are closed under addition and multiplication. If we let q be the integer 3k + 5, then by substitution we have shown 3n + 7 = 2q. Therefore, if n is an odd integer, we have shown that 3n + 7 is even. 

If n is odd, then 3n + 7 is even. Proof: We suppose that n is an odd integer and want to prove that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7 3n + 7 = 6k + 10 3n + 7 = 2(3k + 5). Since k is an integer, 3k + 5 is also an integer because integers are closed under addition and multiplication. If we let q be the integer 3k + 5, then by substitution we have shown 3n + 7 = 2q. Therefore, if n is an odd integer, we have shown that 3n + 7 is even. 

Alternative proof: Proof: We suppose that n is an odd integer and want to prove that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7 = 6k + 10 = 2(3k + 5) = 2q, where q = 3k +5 is an integer because k is an integer, and integers are closed under addition and multiplication. Therefore, we have shown that 3n + 7 is even when n is an odd integer. QED

Writing Guidelines We do not consider a proof complete until there is a well-written proof.

Writing Guidelines We do not consider a proof complete until there is a well-written proof. 0. Do all of the thinking, work, and planning first. A “Know-Show” table or outline or notes or scratch work should be completed prior to writing so you can focus on quality writing, not math.

Writing Guidelines We do not consider a proof complete until there is a well-written proof. 0. Do all of the thinking, work, and planning first. A “Know-Show” table or outline or notes or scratch work should be completed prior to writing so you can focus on quality writing, not math.

1. Begin with a carefully worded statement of the theorem or result to be proven. State what you are about to prove. Below that, write “Proof” and begin writing.

Writing Guidelines We do not consider a proof complete until there is a well-written proof. 0. Do all of the thinking, work, and planning first. A “Know-Show” table or outline or notes or scratch work should be completed prior to writing so you can focus on quality writing, not math.

1. Begin with a carefully worded statement of the theorem or result to be proven. State what you are about to prove. Below that, write “Proof” and begin writing.

2. Begin the proof with a statement of assumptions. “We assume (the hypothesis)...” or “Suppose (the hypothesis)...”

Writing Guidelines We do not consider a proof complete until there is a well-written proof. 0. Do all of the thinking, work, and planning first. A “Know-Show” table or outline or notes or scratch work should be completed prior to writing so you can focus on quality writing, not math.

1. Begin with a carefully worded statement of the theorem or result to be proven. State what you are about to prove. Below that, write “Proof” and begin writing.

2. Begin the proof with a statement of assumptions. “We assume (the hypothesis)...” or “Suppose (the hypothesis)...”

3. Use the pronoun “we.” Mathematicians are a loving community that does everything together. Do not use “I”, “my”, “you” or similar pronouns in writing proofs. It is our convention that we use the pronouns “we” and “our” and “us.”

Writing Guidelines Continued ... 4. Use italics for variables when typing. 5. Display important equations and mathematical expressions. They should be centered and well-aligned. 6. Tell the reader when you are done. Give some form of QED: Quod Erat Demonstrandum - “which was to be demonstrated.” Use whatever symbol you like:  or ⋄ or ♣ or ♡ or $.

Constructing a Proof of a Conditional Statement Proposition: If x and y are odd integers, then x · y is an odd integer.