Question

1. A) List the elements of each of the following suts i) ( $ x: x $ is natural number less than 5 \} ii) (xix is a negative integer greater than - 3) iii) (xix is a positive number less than - 5$ ) \cup(1,2,3) $ iv) (x: $ x $ is a positive even number less tian 10$ ) \cap(x \cdot x $ is an integer. $ ) $ v) $ \{x: x=4 k-1 $, where $ k=0,1,2,3,4,5\} $ vi) $ \{x: x $ is an integer $ \} \cap\{(\ldots \sqrt{2}, 3.14,7\} $ B) Given that $ 1=\{-2,-1,0,1,2\} $. List the eiements of the following sets i) $ \left\{x^{3}: x \in A\right\} $ ii) $ \left\{x^{2}+2: x \leq A\right\} \quad $ iii) $ \{2 / x+1: x \equiv A\} $ iv) $ \left\{3 x^{2}+1: x \in\right. $ 2. Deseribe each of the following in set builder notation a) $ A=\{1,4,2,16,25\} $ b) $ B=\{-7,-5,-3,-1\} $ c) $ C=\{(2,4,6,8,19,12,16\} $ d) $ D=:(1,2,4,3,16,32\} $ 3. Let $ A:=\{1,2,3,4,5\}, B=\{2,4,6\}, B=\{3,4,5\} $ and let $ E=\{0,1,2,3,4,5,6,7,8\} $ Find i) $ B^{\prime} $ ii) $ A \cup B $ iii) $ A \cap B $ ii) $ (A \cup B)^{\prime} $ v) $ (A \cap B)^{\prime} $ vi) $ C-B $ vii) $ (U-A) \cap(B-C)^{+} \quad $ Viii) $ A \cup(C-B) $ \$4. Verify or show the following prope ties: a) Associative Laws: $ (A \cup B) \cup C=A \cup(B \cup) $ and $ (A \cap B) \cap C=A \cap(B \cap C) $ b) Distributive Laws : $ A \cap(B \cup C)=(A \cap B) \cup(A \cap C) $ and $ A \cup(B \cap C)=(A \cup B) \cap(A \cup C) $ 5. i) Prove the De Morgan's Laws a) $ (B \cap C)^{\prime}=B^{\prime} \cup C^{\prime} $ b) $ (B \cup C)^{\prime}=B^{\prime} \cap C^{\prime} $ ii) Prove that $ \left(A^{\prime}\right)^{\prime}=A $ iii) Verify or show the De Morgan's Laws a) $ (B \cap)^{\prime}=B^{\prime} \cup C^{\prime} $ b) $ (B \cup C)^{\prime}=B^{\prime} \cap C^{\prime} $ 6. If $ C \subset D $, then simplify if possible i) $ C \cap D $ ii) $ C^{\prime} \cup D^{\prime} $ iii) $ C \cup D^{\prime} $ iv) $ C^{\prime} r \cdot(C \cup D) $ 7. If $ C $ and $ D $ are disjoint, simplify if possible i) $ C^{\prime} \cap D^{\prime} $ ii) $ C^{\prime} \cup D^{\prime} $ iii) $ (C \cap D)^{\prime} $ iv) $ (C \cup D)^{\prime} $ Represent each of the following on a Yenn diagran

          1. A) List the elements of each of the following suts
i) ( \( x: x \) is natural number less than 5 \}
ii) (xix is a negative integer greater than - 3)
iii) (xix is a positive number less than - 5\( ) \cup(1,2,3) \)
iv) (x: \( x \) is a positive even number less tian 10\( ) \cap(x \cdot x \) is an integer. \( ) \)
v) \( \{x: x=4 k-1 \), where \( k=0,1,2,3,4,5\} \)
vi) \( \{x: x \) is an integer \( \} \cap\{(\ldots \sqrt{2}, 3.14,7\} \)
B) Given that \( 1=\{-2,-1,0,1,2\} \). List the eiements of the following sets
i) \( \left\{x^{3}: x \in A\right\} \)
ii) \( \left\{x^{2}+2: x \leq A\right\} \quad \) iii) \( \{2 / x+1: x \equiv A\} \)
iv) \( \left\{3 x^{2}+1: x \in\right. \)
2. Deseribe each of the following in set builder notation
a) \( A=\{1,4,2,16,25\} \)
b) \( B=\{-7,-5,-3,-1\} \)
c) \( C=\{(2,4,6,8,19,12,16\} \)
d) \( D=:(1,2,4,3,16,32\} \)
3. Let \( A:=\{1,2,3,4,5\}, B=\{2,4,6\}, B=\{3,4,5\} \) and let \( E=\{0,1,2,3,4,5,6,7,8\} \)

Find
i) \( B^{\prime} \)
ii) \( A \cup B \)
iii) \( A \cap B \)
ii) \( (A \cup B)^{\prime} \)
v) \( (A \cap B)^{\prime} \)
vi) \( C-B \)
vii) \( (U-A) \cap(B-C)^{+} \quad \) Viii) \( A \cup(C-B) \)
\$4. Verify or show the following prope ties:
a) Associative Laws: \( (A \cup B) \cup C=A \cup(B \cup) \) and \( (A \cap B) \cap C=A \cap(B \cap C) \)
b) Distributive Laws : \( A \cap(B \cup C)=(A \cap B) \cup(A \cap C) \) and \( A \cup(B \cap C)=(A \cup B) \cap(A \cup C) \)
5. i) Prove the De Morgan's Laws
a) \( (B \cap C)^{\prime}=B^{\prime} \cup C^{\prime} \)
b) \( (B \cup C)^{\prime}=B^{\prime} \cap C^{\prime} \)
ii) Prove that \( \left(A^{\prime}\right)^{\prime}=A \)
iii) Verify or show the De Morgan's Laws a) \( (B \cap)^{\prime}=B^{\prime} \cup C^{\prime} \)
b) \( (B \cup C)^{\prime}=B^{\prime} \cap C^{\prime} \)
6. If \( C \subset D \), then simplify if possible
i) \( C \cap D \)
ii) \( C^{\prime} \cup D^{\prime} \)
iii) \( C \cup D^{\prime} \)
iv) \( C^{\prime} r \cdot(C \cup D) \)
7. If \( C \) and \( D \) are disjoint, simplify if possible
i) \( C^{\prime} \cap D^{\prime} \)
ii) \( C^{\prime} \cup D^{\prime} \)
iii) \( (C \cap D)^{\prime} \)
iv) \( (C \cup D)^{\prime} \)

Represent each of the following on a Yenn diagran

$1. A) List the elements of each of the following suts i) ( x: x is natural number less than 5 } ii) (xix is a negative integer greater than - 3) iii) (xix is a positive number less than - 5) ∪(1,2,3) iv) (x: x is a positive even number less tian 10) ∩(x · x is an integer. ) v) {x: x=4 k-1, where k=0,1,2,3,4,5} vi) {x: x is an integer }∩{(…√(2), 3.14,7} B) Given that 1={-2,-1,0,1,2}. List the eiements of the following sets i) {x^3: x ∈ A} ii) {x^2+2: x ≤ A} iii) {2 / x+1: x ≡ A} iv) {3 x^2+1: x ∈. 2. Deseribe each of the following in set builder notation a) A={1,4,2,16,25} b) B={-7,-5,-3,-1} c) C={(2,4,6,8,19,12,16} d) D=:(1,2,4,3,16,32} 3. Let A:={1,2,3,4,5}, B={2,4,6}, B={3,4,5} and let E={0,1,2,3,4,5,6,7,8} Find i) B^' ii) A ∪ B iii) A ∩ B ii) (A ∪ B)^' v) (A ∩ B)^' vi) C-B vii) (U-A) ∩(B-C)^+ Viii) A ∪(C-B) $4. Verify or show the following prope ties: a) Associative Laws: (A ∪ B) ∪ C=A ∪(B ∪) and (A ∩ B) ∩ C=A ∩(B ∩ C) b) Distributive Laws : A ∩(B ∪ C)=(A ∩ B) ∪(A ∩ C) and A ∪(B ∩ C)=(A ∪ B) ∩(A ∪ C) 5. i) Prove the De Morgan's Laws a) (B ∩ C)^'=B^'∪ C^' b) (B ∪ C)^'=B^'∩ C^' ii) Prove that (A^')^'=A iii) Verify or show the De Morgan's Laws a) (B ∩)^'=B^'∪ C^' b) (B ∪ C)^'=B^'∩ C^' 6. If C ⊂ D, then simplify if possible i) C ∩ D ii) C^'∪ D^' iii) C ∪ D^' iv) C^' r ·(C ∪ D) 7. If C and D are disjoint, simplify if possible i) C^'∩ D^' ii) C^'∪ D^' iii) (C ∩ D)^' iv) (C ∪ D)^' Represent each of the following on a Yenn diagran$

Added by Lisa B.

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