**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}

**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}

**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’**

**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

**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’

**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}

**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}

**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}

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