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Jump to level 1 Select each step to complete a proof by contrapositive of the theorem below. Theorem: For every pair of integers x and y, if 5xy + 8 is even, then at least one of x or y must be even. 1. Let x and y be integers. We will assume that it is not true that x or y is even and will show that 5xy + 8 is odd. 2 Select Select 3: If it is not true that x or y is even, then x and y are both odd. Therefore 5xy + 8 = 5(2k + 1)(2) + 1) +8. x=2k + 1 and y=2) + 1 for some integers k and J. Select Select 7. Since 5xy + 8 is equal to two times an integer plus 1, 5xy + 8 is odd. 4 5: 6: Check Next 3 Feedba
CHALLENGE 2.8.1: Proof by contrapositive. ACTIVITY Cate Jump to level 1 Select each step to complete a proof by contrapositive of the theorem below. Theorem: For every pair of integers x and y, if 5xy +8 is even, then at least one of x or y must be 11 Let x and y be integers. We will assume that it is not true that x or y is even and will show TA) Select Select Select 4 If it is not true that x or y is even, then x and y are both odd. Therefore 5xy + 8 = 5(2k + 1)(2j+1) + 8. 5: x = 2k +1 and y=2j+1 for some integers k and J. 6: Select 7. Since 5xy + 8 is equal to two times an integer plus 1, 5xy +8 is odd. 2 3. Check Additional exercises Next Prove each statement by contrapositive H EXERCISE 2.8.1 Proof by contrapositive of statements about oda and even integers.
CHALLENGE ACTIVITY 2.8.1: Proof by contrapositive. 413774.252714227 Jump to level 1 Select each step to complete a proof by contrapositive of the theorem below. Theorem: For every pair of integers x and y. if 5xy + 8 is even, then at least one of x or y must be ev 1. Let x and y be integers. We will assume that it is not true that x or y is even and will show ti 2: Select 3: Select 4: 5 6: 7. Select to Select 5(2k + 1)(2) + 1) + 8 = 20jk + 10j + 10k + 5+8=2(10jk + 5j + 5k + 6) +1 Since j and k are both integers, 10jk + 5j + 5k + 6 is also an integer. Therefore 5xy + 8 = 5(2k + 1)(2j+ 1) + 8. x = 2k + 1 and y = 2j+ 1 for some integers k and j. Since 5xy + 8 is equal to two times an integer plus 1, 5xy +8 is odd
4137742527142 qx3zy7 Jump to level 1 Select each step to complete a proof by contrapositive of the theorem below. Theorem: For every pair of integers x and y, if 5xy + 8 is even, then at least one of x or y must 1. Let x and y be integers. We will assume that it is not true that x or y is even and will sh Select 3: Select 4: Select 5: Select Select 6: 5(2k + 1)(2) + 1) +8=20jk + 10j + 10k + 5 + 8 = 2(10jk + 5j + 5k + 6) + 1 Since j and k are both integers, 10jk + 5j + 5k + 6 is also an integer. Therefore 5xy + 8 = 5(2k + 1)(2) + 1) + 8. 7. ■ x = 2k + 1 and y=2j + 1 for some integers k and j. 2: Check to Additional exercises Next 2
4137742 Jump to level 1 Select each step to complete a proof by contrapositive of the theorem below. Theorem: For every pair of integers x and y, if 5xy + 8 is even, then at least one of x or y must be ever 1. Let x and y be integers. We will assume that it is not true that x or y is even and will show tha 2: Select 3: Select 4: Select 5: Select 6: Select Select 7. ■ Check A 5(2k+1)(2)+1)+8=20jk + 10j + 10k +5+8=2(10jk + 5j + 5k + 6) +1 Since j and k are both integers, 10jk + 5j + 5k + 6 is also an integer. Therefore 5xy + 8 = 5(2k + 1)(2) + 1) +8. x=2k+1 and y=2j+ 1 for some integers k and j. Additional exercises V Next sporitive of statements about odd and even integers.