Let U be the universal set and A and B be two subsets of U. In each case, we describe the region that should be shaded in the Venn diagram.
(i) (A ∪ B)'
The set \( A \cup B \) represents all elements which are in A or in B or in both.
Therefore, \( (A \cup B)' \) represents all elements of the universal set U which are neither in A nor in B.
Venn diagram description:
Shade the region outside both circles A and B, but inside the universal set U.
(ii) A' ∩ B'
\( A' \) represents all elements not in A, and \( B' \) represents all elements not in B.
Their intersection \( A' \cap B' \) consists of elements which are neither in A nor in B.
Venn diagram description:
Shade the region outside both A and B circles.
(iii) (A ∩ B)'
The set \( A \cap B \) represents elements common to both A and B.
Therefore, \( (A \cap B)' \) represents all elements of U except the common region of A and B.
Venn diagram description:
Shade the entire universal set except the overlapping (common) region of A and B.
(iv) A' ∪ B'
\( A' \) contains elements not in A, and \( B' \) contains elements not in B.
Their union \( A' \cup B' \) consists of all elements which are not common to both A and B.
By De Morgan’s law, \[ A' \cup B' = (A \cap B)'. \]
Venn diagram description:
Shade the entire region except the intersection of A and B.
Thus, the shaded regions in (i) and (ii) are the same, and the shaded regions in (iii) and (iv) are the same.
Let f: ℝ → ℝ be defined as
\(f(x) = \left\{ \begin{array}{ll} [e^x] & x < 0 \\ [a e^x + [x-1]] & 0 \leq x < 1 \\ [b + [\sin(\pi x)]] & 1 \leq x < 2 \\ [[e^{-x}] - c] & x \geq 2 \\ \end{array} \right.\)
Where a, b, c ∈ ℝ and [t] denotes greatest integer less than or equal to t.
Then, which of the following statements is true?