1.1 Sets-Q&A Exercises-Elements of Modern Algebra 8th Edition

EXERCISE 1.1 SETS
1.


True
True
False
True
True
False
True
True
False
False




𝐴 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑛𝑜𝑛𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑒𝑣𝑒𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 12} 𝐴 = {𝑥||𝑥| = 1}

𝐴 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟} 𝐴 = {𝑥|𝑥 = 𝑘 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑘 ∈ 𝒁 }

False True
The corrected: {2} ⊆ 𝐴 Since {11,2,7} = {2,7,11}
False False
The corrected: {2,11} = 𝐴 ∩ 𝐵 The corrected: 7 ∈ 𝐴 𝑎𝑛𝑑 11 ∈ 𝐴
False True
The corrected: 𝐴 ⊈ 𝐵 𝑠𝑖𝑛𝑐𝑒 7 ∈ 𝐴 𝑏𝑢𝑡 7 ∉ 𝐵 Since {7,11,2} = {2,7,11}

True True

True True

True False
∅ is a subset of every set. ∅ has no element.
True True
∅ is a subset of every set.
False False
∅ is a subset of every set but not vice versa. {∅} has an element but ∅ has no element.
False True
∅ has no element. ∅ is a subset of every set.

False True
True False
True False
False 𝐴∩∅=∅≠𝐴=𝐴∪∅
𝐴 ∩ (𝐵 ∪ 𝐶) = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶) True
𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶) De Morgan's Law: 𝐵 ∩ 𝐶 = (𝐵 ∪ 𝐶)′
False False
(𝐴 ∪ 𝐵) ∩ 𝐶 = (𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶) (𝐴 ∩ 𝐵) ∪ 𝐶 = (𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶)
False False
𝐴 ∩ (𝐵 ∪ 𝐶) = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶)

, 2.

= {0,1,2,3,4,5,6,8,10} = {2,3,5}
= {0,2,4,6,7,8,9,10} = 𝐴 ∩ (𝐵 ∩ 𝐶)
= 𝐴 ∩ (𝐵 ∩ 𝐶) = {2}
=∅ = {0,1,2,3,4,5}
= {0,2,3,4,5} = {6,8,10}
𝐴 = {6,7,8,9,10} = {1,3,5} = {6,8,10}
𝐵 ∩ 𝐶 = {2} = {1,2,3,5} = {2,3,5,7}
𝐵 ∪ 𝐶 = {0,2,3,4,5,6,7,8,10} = {3,5} = {1}
𝐵 − 𝐶 = {0,4,6,8,10}
𝐵 = {1,3,5,7,9}
𝐴 ∪ 𝐵 = {0,1,2,3,4,5,7,9}
𝐶 − 𝐵 = {3,5,7}
𝐴 − 𝐶 = {0,1,4}


=𝐴 =𝐴 =∅
=𝑈 =𝐴 =∅
=𝐴 =𝑈 =𝑈
=𝐴 =𝑈 =∅
=𝐴 =∅


𝒫(𝐴) = ∅, {𝑎} 𝒫(𝐴) = ∅, {0}, {1}, {0,1}

∅, {𝑎}, {𝑏}, {𝑐}, {𝑎, 𝑏} ∅, {1}, {2}, {3}, {4}, {1,2}, {1,3}
𝒫(𝐴) =
, {𝑎, 𝑐}, {𝑏, 𝑐}, {𝑎, 𝑏, 𝑐} 𝒫(𝐴) = , {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}
, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}
𝒫(𝐴) = ∅, {1}, {1} , 1, {1}
𝒫(𝐴) = ∅, {1}
𝒫(𝐴) = ∅, {∅}

𝒫(𝐴) = ∅, {∅}, {∅} , ∅, {∅}



- 𝑋 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟}, 𝑋 = - 𝑋 = {𝑎, 𝑏}, 𝑋 = {𝑐, 𝑑}
{0}, 𝑋 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟} - 𝑋 = {𝑎}, 𝑋 = {𝑏, 𝑐}, 𝑋 = {𝑑}
- 𝑋 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟}, 𝑋 =
{𝑥|𝑥 𝑖𝑠 𝑎 𝑛𝑜𝑛 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟} - 𝑋 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟}, 𝑋 =
{𝑥|𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑛𝑢𝑚𝑏𝑒𝑟}
- 𝑋 = {1,5,9}, 𝑋 = {11,15} - 𝑋 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟}, 𝑋 =
- 𝑋 = {1,5}, 𝑋 = {9,11}, 𝑋 = {15} {𝑥|𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟}, 𝑋 =
{𝑥|𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑛𝑢𝑚𝑏𝑒𝑟}


- 𝑋 = {1}, 𝑋 = {2}, 𝑋 = {3} - 𝑋 = {1}, 𝑋 = {2,3,4}
- 𝑋 = {1}, 𝑋 = {2,3} - 𝑋 = {2}, 𝑋 = {1,3,4}
- 𝑋 = {2}, 𝑋 = {1,3} - 𝑋 = {3}, 𝑋 = {1,2,4}
- 𝑋 = {3}, 𝑋 = {1,2} - 𝑋 = {4}, 𝑋 = {1,2,3}
- 𝑋 = {1}, 𝑋 = {2}, 𝑋 = {3,4}
- 𝑋 = {1}, 𝑋 = {3}, 𝑋 = {2,4}
- 𝑋 = {1}, 𝑋 = {4}, 𝑋 = {2,3}
- 𝑋 = {2}, 𝑋 = {3}, 𝑋 = {1,4}
- 𝑋 = {2}, 𝑋 = {4}, 𝑋 = {1,3}
- 𝑋 = {3}, 𝑋 = {4}, 𝑋 = {1,2}
- 𝑋 = {1}, 𝑋 = {2}, 𝑋 = {3}, 𝑋 = {4}