1. If A’ = {1, 2, 3, 4}, U = {1, 2, 3, 4, 5, 6, 7, 8}, find A in U and draw Venn diagram

Solution:

 A’ = {5, 6, 7, 8}

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2. If U = {x/x € 25, x€N}. A = {x/x € U, x ≤ 15} and B = {x/x € U, 0 < x ≤ 25}, list the elements of the following sets and draw Venn diagram:

(i) A’ in U:

(ii) B’ in U

(iii) A\B;

(iv) A Δ B

Solution:

U = {1, 2, 3, 4 ……….25}

A = {1, 2, 3, 4……….15}

B = {1, 2, 3 ….25}

(i) A’ = {16, 17, 18, 19….25}

 

(ii) B’ = { }

(iii) A\B = { }

(iv) A Δ B = A \ B U B \ A

= { } U {16, 17, 18… 25}

= {16, 17, 18 …..25}

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3. Let A and B subsets of a set U. Identify the wrong statements:

(i) (A’)’ = A

(ii) A \ B = B \ A

(iii) A U A’ = U

(iv) A Δ B = B Δ A

(v) (A \ B)’ = A’ \ B’

Solution:

 If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8, 9}

(i) (A’)’ = {2, 4, 6, 8}

(A’)’ = {1, 3, 5, 7, 9} = A

(A’)’ = A

 

(ii) A \ B = {1, 3, 5, 7}

B \ A = {2, 4, 6, 8}

We see that A \ B ≠ B \ A

(iii) A U A’ = {1, 3, 5, 7, 9} U {2, 4, 6, 8}

= {1, 2, 3, 4, 5, 6, 7, 8, 9}

= U

A U A’ = U

(iv) A Δ B =(A \ B) U (B \ A)

= {1, 3, 5, 7} U {2, 4, 6, 8}

= {1, 2, 3, 4, 5, 6, 7, 8}

B Δ A = (B \ A) U (A \ B)

= {2, 4, 6, 8} U {1, 3, 5, 7}

= {1, 2, 3, 4, 5, 6, 7, 8}

A Δ B = B Δ A

 

(v) A \ B = {1, 3, 5, 7}

(A \ B)’ = {2, 4, 6, 8, 9}

A’ = {2, 4, 6, 8} and B’ = {1, 3, 5, 7}

A’ \ B’ = {2, 4, 6, 8}

Hence (A \ B)’ ≠ A’ \ B’

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4. Suppose U = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}. A = {3, 4, 5, 6, 9}, B = {3, 7, 9, 5} and C = {6, 8, 10, 12, 7}. Write down the following sets and draw Venn diagram for each:

(i) A’

(ii) B’

(iii) C’

(iv) (A’)’

(v) (B’)’

(iv) (C’)’

Solution:

(i) A’ = {7, 8, 10, 11, 12, 13}

(ii) B’ = {4, 6, 8, 10, 11, 12, 13}

(iii) C’ = {3, 4, 5, 9, 11, 13}

(iv) A’ = {7, 8, 10, 11, 12, 13}

(A’)’ = {3, 4, 5, 6, 9} = A

(v) (B’)’ = B’ = {4, 6, 8, 10, 11, 12, 13}

(B’)’ = {3, 7, 9, 5} = B

(vi) (C’)’ = {3, 4, 5, 9, 11, 13}
(C’)’ = {6, 8, 10, 12, 7} = C

 

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5. Suppose U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4} and B = {2, 4, 6, 8}, write down the following sets and draw Venn diagram.
(i) A’

(ii) B’

(iii) A U B

(iv) A ∩ B (v) (A U B)’ (vi) (A ∩B)’
How (A UB)’ is related to A’ and B’? What relation you see between
(A ∩ B)’ and A’ and B’
Solution:

(i) A’ = {5, 6, 7, 8, 9}

(ii) B’ = {1, 3, 5, 7, 9}

(iii) A U B = {1, 2, 3, 4, 6, 8}

(iv) A ∩ B = {2, 4}

(v) (A UB)’
(A U B) = {1, 2, 3, 4, 6, 8}
(A U B)’ = {5, 7, 9}

vi) (A ∩ B)’
(A ∩ B) = {2, 4}
(A ∩ B)’ = {1, 3, 5, 6, 7, 8}

 

We see that (A U B)’ = A’ ∩ B’
(A ∩ B)’ = A’ U B’

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6. Find (A \ B) and (B \ A) for the following sets and draw Venn diagram.
(i) A = {a, b, c, d, e, f, g, h} and
B = {a, e, i, o, u}
(ii) A = {1, 2, 3, 4, 5, 6} and
B = {2, 3, 5, 7, 9}
(iii) A = {1, 4, 9, 16, 25} and
B = {1, 2, 3, 4, 5, 6, 7, 8, 9}
(iv) A = {x | x is a prime number less than 5} and
B = {x | x is a square number less than 16}
Solution:
(i) A = { a, b, c, d, e, f, g, h}
B = {a, e, i, o, u}
A \ B = {b, c, d, f, g, h}

B \ A = {i, o, u}

(ii) A = {1, 2, 3, 4, 5, 6} and
B = {2, 3, 5, 7, 9}
A \ B = {1, 4, 6}

B \ A = {7, 9}

(iii) A = {1, 4, 9, 16, 25} and
B = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A \ B = {16, 25}

B \ A = {2, 3, 5, 6, 7, 8}

(iv) A = {x | x is a prime number less than 5}
= {2, 3}
B = {x | x is a square number less than 16}
= {1, 4, 9}
A \ B = {2, 3}

B \ A = {1, 4, 9}

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7. Looking at the Venn diagram list the elements of the following sets:
(i) A \ B
(ii) B \ A
(iii) A \ C
(iv) C \ A
(v) B \ C
(vi) C \ B

Solution:
(i) A \ B = {1, 2, 7}
(ii) B \ A = {5, 6}
(iii) A \ C = {1, 2, 3}
(iv) C \ A = {6, 8, 9}
(v) B \ C = {5, 3}
(vi) C \ B = {7, 8, 9}

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8. Find A Δ B and draw Venn diagram when:
(i) A = {a, b, c, d} and B = {d, e, f}
(ii) A = {1, 2, 3, 4, 5} and B = {2, 4}
(iii) A ={1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}
(iv) A = {1, 4, 7, 8} and B = {4, 8, 6, 9}
(v) A = {a, b, c, d, e} and B = {1, 3, 5, 7}
(vi) A = {1, 2, 3, 4, 5} and B = {1, 3, 5, 7}
Ans:
(i) A = {a, b, c, d} B = {d, e, f}
A \ B = {a, b, c}
B \ A = {e, f}
A Δ B = {a, b, c, e, f}

(ii) A = {1, 2, 3, 4, 5}                            B = {2, 4}
A \ B = {1, 3, 5}
B \ A = { }
A Δ B = {1, 3, 5}

(iii) A ={1, 2, 3, 4, 5} ;          B = {1, 2, 3, 4, 5, 6}
A \ B = {.}
B \ A = {6}
A Δ B = {6}

(iv) A = {1, 4, 7, 8}; B = {4, 8, 6, 9}
A \ B = {1, 7]
B \ A = {6, 9}
A Δ B = {1, 6, 7, 9}

(v) A = {a, b, c, d, e} and B = {1, 3, 5, 7}
A \ B = {b, d}
B \ A = {g}
A Δ B = {b, d, g}

(vi) A = {1, 2, 3, 4, 5} and B = {1, 3, 5, 7}
A \ B = {2, 4}
B \ A = {7}
A Δ B = {2, 4, 7}