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Question
Show that the following four conditions are equivalent: (i) A ⊂ B (ii) A – B = Φ (iii) A ∪ B = B (iv) A ∩ B = A
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Solution
The sets are given as,
A ⊂ B (i)
A − B = Φ (ii)
A ∪ B = B (iii)
A ∩ B = A (iv)
Venn diagram of A ⊂ B is shown below:
Figure (1)
It means all the elements of A belongs to B .
Condition (ii) means A has no element different from B
Condition (iii) means all the elements either in A or B are in B .
Condition (iv) means common elements of A and B are the elements of A
Hence, conditions (i), (ii), (iii) and (iv) are equivalent.
The sets are given as,
Venn diagram of
Figure (1)
It means all the elements of
Condition (ii) means
Condition (iii) means all the elements either in
Condition (iv) means common elements of
Hence, conditions (i), (ii), (iii) and (iv) are equivalent.
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