Question:
Let $Z$ be the set of integers. If $A \, = \, \{ x \in Z : 2^{ (x+2) (x^2 - 5x + 6)} \}=1$ and $B \, = \, \{ \, x \in Z: -3 <2x -1 <9 \}$, then the number of subsets of the set $A \times B$, is :
Let $Z$ be the set of integers. If $A \, = \, \{ x \in Z : 2^{ (x+2) (x^2 - 5x + 6)} \}=1$ and $B \, = \, \{ \, x \in Z: -3 <2x -1 <9 \}$, then the number of subsets of the set $A \times B$, is :
Updated On: Aug 19, 2024
- $2^{18}$
- $2^{10}$
- $2^{15}$
- $2^{12}$
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The Correct Option is C
Solution and Explanation
\(A \, = \, \{ \, x \in z : 2 ^{(x+2)(x^2-5x+6)} \, = \, 1\}\)
\(2^{(x+2)(x^2-5x+6)} \, = \, 2^{0} \, \Rightarrow \, x \, = -2,2,3\)
\(A \, = \, \{ -2, 2, 3\}\)
\(B \, = \, \, {x \in Z : -3 <2x - 1 < 9}\)
\(A \, \times \, B=3\times15=15\) has \(15\) elements so number of subsets
of \(A \times B\) is \(2^{15}\)
\(2^{(x+2)(x^2-5x+6)} \, = \, 2^{0} \, \Rightarrow \, x \, = -2,2,3\)
\(A \, = \, \{ -2, 2, 3\}\)
\(B \, = \, \, {x \in Z : -3 <2x - 1 < 9}\)
\(A \, \times \, B=3\times15=15\) has \(15\) elements so number of subsets
of \(A \times B\) is \(2^{15}\)
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".