Question 1.

Using the given venn diagram, write the elements of

(i) A

(ii) B

(iii) A ∪ B

(iv) A ∩ B

(v) A – B

(vi) B – A

(vii) A’

(viii) B’

(ix) U

Solution:

(i) A = {2, 4, 7, 8, 10}

(ii) B = {3, 4, 6, 7, 9, 11}

(iii) A ∪ B = {2, 3, 4, 6, 7, 8, 9, 10, 11}

(iv) A ∩ B = {4, 7}

(v) A – B = {2, 8, 10}

(vi) B – A = {3, 6, 9, 11}

(vii) A’ = {1, 3, 6, 9, 11, 12}

(viii) B’ = {1, 2, 8, 10, 12}

(ix) U = {1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12}.

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Question 2.

Find A ∪ B, A ∩ B, A – B and B – A for the following sets.

(i) A = {2, 6, 10, 14} and B = {2, 5, 14, 16}

(ii) A = {a, b, c, e, u} and B = {a, e, i, o, u]

(iii) A = {x : x ∈ N, x ≤ 10} and B = {x : x ∈ W, x < 6}

(iv) A = Set of all letters in the word “mathematics” and B = Set of all letters in the word “geometry”

Solution:

(i) A = {2, 6, 10, 14} and B = {2, 5, 14, 16}

A ∪ B = {2, 6, 10, 14} ∪ {2, 5, 14, 16} = {2, 5, 6, 10, 14, 16}

A ∩ B = {2, 6, 10, 14} ∩ {2, 5, 14, 16} = {2, 14}

A – B = {2, 6, 10, 14} – {2, 5, 14, 16} = {6, 10}

B – A = {2, 5, 14, 16} – {2, 6, 10, 14} = {5, 16}

(ii) A = {a, b, c, e, u} and B = {a, e, i, o, u}

A ∪ B = {a, b, c, e, u) ∪ {a, e, i, o, u) = {a, b, c, e, i, o, u}

A ∩ B = {a, b, c, e, u} ∩ {a, e, i, o, u} {a, e, u}

A – B = {a, b, c, e, u) – {a, e, i, o, u) = {b, c}

B – A = {a, e, i, o, u} – {a, b, c, e, u} = {i, o}

(iii) x ∈ {1, 2, 3, ……..} ; x ∈ {0, 1, 2, 3, 4, 5, ……..}

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

B = {0, 1, 2, 3, 4, 5}

A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ∪ {0, 1, 2, 3, 4, 5} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A ∩ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ∩ {0, 1, 2, 3, 4, 5} = {1, 2, 3, 4, 5}

A – B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {0, 1, 2, 3, 4, 5} = {6, 7, 8, 9, 10}

B – A = {0, 1, 2, 3, 4, 5} – {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} = {0}

(iv) A= {m, a, t, h, e, i, c, s), B = {g, e, o, m, t, r, y)

A ∪ B = {m ,a, t, h, e, i, c, s} ∪ {g, e, o, m, t, r, y} = {m, a, t, h, e, i, c, s, g, o, r, y)

A ∩ B = {m, a, t, h, e, i, c, s} ∩ {g, e, o, m, t,r,y} = {m, t, e}

A – B = {m ,a, t, h, e, i, c, s} ∪ {g, e, o, m, t, r, y} = {a, h, i, c, s)

B – A = {m, a, t, h, e, i, c, 5} ∩ {g, e, o, m, t,r,y} = {g, o, r, y}

Question 3.

If U = {a, b, c, d, e,f g ,h}, A = {b, d, f, h} and B = {a, d, e, h}, find the following sets.

(i) A’

(ii) B’

(iii) A’ ∪ B’

(iv) A’ ∩ B’

(v) (A ∪ B)’

(vi) (A ∩ B)’

(vii) (A’)’

(viii) (B’)’

Solution:

(i) A’ = U – A = {a, b, c, d, e, f, g, y} – {b, d, f, h} = {a, c, e, g}

(ii) B’ = U – B = {a, b, c, d, e, f, g, y) – {a, d, e, h] = {b, c, f, g}

(iii) A’ ∪ B’= {a, c, e, g} ∪ {b, c, f, g} = {a, b, c, e, f g}

(iv) A’ ∩ B’= {a, c, e, g} ∩ {b, c, f, g} = {c, g}

(v) (A ∪ B)’ = U – (A ∪ B) = {a, b, c, d, e, f, g, y) – {a, b, d, e, f, h} = {c, g}

(vi) (A ∩ B)’ = U – (A ∩B) = {a, b, c, d, e, f, g, y} – {d, h} = {a, b, c, e, f, g}

(vii) (A’)’ = U – A’ = {a, b, c, d, e, f, g, h} – {a, c, e, g} = {b, d, f, h)

(viii) (B’)’ = U – B’ = {a, b, c, d, e, f, g, h} – {b, c, f, g} = {a, d, e, h}

Question 4.

Let U = {0, 1, 2 , 3, 4, 5, 6, 7}, A = {1, 3, 5, 7} and B = {0, 2, 3, 5, 7}, find the following sets.

(i) A’

(ii) B’

(iii) A ‘ ∪ B’

(iv) A’ ∩ B’

(v) (A ∪ B)’

(vi) (A ∩ B)’

(vii) (A’)’

(viii) (B’)’

Solution:

(i) A’ = U – A = {0, 1 ,2, y, 4, 5, 6, 7} – {1, 3, 5, 7} = {0, 2, 4, 6}

(ii) B’ = U – B = {0, 1, 2, 3, 4, 5, 6 ,7} – {0, 2, 3, 5, 7} = {1, 4, 6}

(iii) A’ ∪ B’ = {0, 2, 4, 6} ∪ {1, 4, 6} = {0, 1, 2, 4, 6}

(iv) A’ ∩ B’ = {0, 2, 4, 6} ∩ {1, 4, 6} = {4, 6}

(v) (A ∪ B)’ = U – (A ∪ B) = {0, 1, 2, 3, 4, 5, 6, 7} – {0, 1, 2, 3, 5, 7} = {4, 6}

(vi) (A ∩ B)’ = U – (A ∩ B)= {0, 1, 2, 3, 4, 5, 6, 7} – {3,5,7} = {0, 1, 2, 4, 6}

(vii) (A’)’ = U – A’ = {0, 1, 2, 3, 4, 5, 6, 7} – {0, 2, 4, 6} = {1, 3, 5, 7}

(viii) (B’)’ = U – B’ = {0, 1, 2, 3, 4, 5, 6, 7} – {1, 4, 6} = {0, 2, 3, 5, 7}.

Question 5.

Find the symmetric difference between the following sets.

(i) P = {2, 3, 5, 7, 11} and Q = {1, 3, 5, 11}

(ii) R = {l, m, n, o, p} and S = {j, l, n, q)

(iii) X = {5, 6, 7} and Y = {5, 7, 9, 10}

Solution:

(i) P = {2, 3, 5, 7, 11}

Q= {1, 3, 5, 11}

P – Q = {2, 3, 5, 7, 11} – {1, 3, 5, 11} = {2, 7}

Q – P = {1, 3, 5, 11} – {2, 3, 5, 7, 11} = {1}

P ∆ Q = (P – Q) ∪ (Q – P) = {2, 7} ∪ {1} = {1, 2, 7}

(ii) R = {l, m, n, o, p}

S = {j, l, n, q}

R – S = {l, m, n, o, p) – {j, l, n, q} = {m, o, p)

s – R = {j, l, n, q) – {l, m, n, o, p}= {j, q}

R ∆ S = (R – S) ∪ (S – R) = {m, o, p) ∪ {j, q} = {j, m, o, p, q)

(iii) X = {5, 6, 7}

Y = {5, 7, 9, 10}

X – Y = {5, 6, 7} – {5, 7, 9, 10} – {6}

Y – X = {5, 6, 9, 10} – {5, 6, 7} = {9, 10}

X ∆ Y = (X – Y) ∪ (Y – X) = {6} ∪ {9, 10} = {6, 9, 10}.

Question 6.

Using the set symbols, write down the expressions for the shaded region in the following

(i)

(ii)

(iii)

Solution:

(i) X – Y

(ii) (X ∪ Y)’

(iii) (X – Y) ∪ (X – Y)

Question 7.

Let A and B be two overlapping sets and the universal set U. Draw appropriate Venn diagram for each of the following,

(i) A ∪ B

(ii) A ∩ B

(iii) (A ∩ B)’

(iv) (B – A)’

(v) A’ ∪ B’

(vi) A’ ∩ B’

(vii)What do you observe from the diagram (iii) and (v)?

Solution:

(i) A ∪ B

(ii) A ∩ B

(iii) (A ∩ B)’

(iv) (B – A)’

(v) A’ ∪ B’

(vi) A’ ∩ B’

(vii) From the diagram (iii) and (v) we observe that (A ∩ B)’ = A’ ∪ B’.