State whether the following statement is true or false. If the statement is false,rewrite it correctly.
If $A$ and $B$ are non-empty sets,then $A \times B$ is a non-empty set of ordered pairs $(x, y)$ such that $x \in A$ and $y \in B$.
If $A$ and $B$ are non-empty sets,then $A \times B$ is a non-empty set of ordered pairs $(x, y)$ such that $x \in A$ and $y \in B$.
(A) The statement is True.
By the definition of the Cartesian product of two sets $A$ and $B$,$A \times B = \{(x, y) : x \in A \text{ and } y \in B\}$. If $A$ and $B$ are non-empty,there exists at least one element $a \in A$ and $b \in B$,which implies $(a, b) \in A \times B$. Thus,$A \times B$ is a non-empty set.
By the definition of the Cartesian product of two sets $A$ and $B$,$A \times B = \{(x, y) : x \in A \text{ and } y \in B\}$. If $A$ and $B$ are non-empty,there exists at least one element $a \in A$ and $b \in B$,which implies $(a, b) \in A \times B$. Thus,$A \times B$ is a non-empty set.
Explore More
Similar Questions
If $(1, 3), (2, 5)$ and $(3, 3)$ are three elements of $A \times B$ and the total number of elements in $A \times B$ is $6$,then the remaining elements of $A \times B$ are
Easy
View SolutionIf $P = \{1, 2\}$,form the set $P \times P \times P$.
Easy
View SolutionIf $A = \{2, 3, 5\}$ and $B = \{2, 5, 6\}$,then find $(A - B) \times (A \cap B)$.
Easy
View SolutionLet $A = \{a, b, c\}$ and $B = \{1, 2\}$. Consider a relation $R$ defined from set $A$ to set $B$. Then $R$ is a subset of which of the following sets?
Medium
View SolutionState whether the following statement is true or false. If the statement is false,rewrite it correctly.
If $A = \{1, 2\}$ and $B = \{3, 4\}$,then $A \times \{B \cap \varnothing\} = \varnothing$.
If $A = \{1, 2\}$ and $B = \{3, 4\}$,then $A \times \{B \cap \varnothing\} = \varnothing$.
Easy
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo