Property 1
Equality of Ordered Pairs
For a and b elements, the (a, b) element is called an ordered pair. a is called the first component, and b is called the second component.
\((a,b)=(c,d)\Leftrightarrow a=c\text{ and }b=d\)
1.
\(x,y\in\mathbb{N}\)
\((x^2-4,y+2)=(5,3)\)
\(\Rightarrow x\cdot y=?\)
2.
\((2^a,3^b)=(4,27)\)
\(\Rightarrow a+b=?\)
3.
\((x+y,y-4)=(10,6)\)
\(\Rightarrow x\cdot y=?\)
4.
\(x,y\in\mathbb{Z}^+\)
\((\sqrt{x},y^2)=(4,9)\)
\(\Rightarrow x\cdot y=?\)
5.
\((2x,y-4,z)=(8,x,6)\)
\(\Rightarrow x+y+z=?\)
6.
\((x+1,y-1,z-3)=(4,x,y)\)
\(\Rightarrow z=?\)
7.
\((x,x\cdot y,y+3)=(4,8,z)\)
\(\Rightarrow z=?\)
Property 2
Let A and B sets.
\(A\times B=\{(x,y)\mid x\in A\text{ and }y\in B\}\) set is called the cartesian product of A and B.
Example
\(\left.\begin{aligned}A=\{1,2\}\\B=\{x,y\}\end{aligned}\right\}\)
\(\Rightarrow A\times B=\{(1,x),(1,y),(2,x),(2,y)\}\)
- \(n(A\times B)=n(A)\cdot n(B)\)
- \(A\times B\ne B\times A\)
- \(A\times(B\cap C)=(A\times B)\cap(A\times C)\)
- \(A\times(B\cup C)=(A\times B)\cup(A\times C)\)
- \(A\times(B\times C)=(A\times B)\times C\)
- \(A\times(B\setminus C)=(A\times B)\setminus(A\times C)\)
1.
\(A=\{1,2\}\)
\(B=\{a,b\}\)
\(\Rightarrow A\times B=?\)
2.
\(A=\{1,3,5\}\)
\(B=\{a,b,c\}\)
\(\Rightarrow n(A\times B)=?\)
3.
\(A\times B=\{(1,a),(1,b),(2,a),(2,b)\}\)
\(\Rightarrow B=?\)
4.
\(A=\{1,2,3\}\)
\(B=\{a,b\}\)
\(C=\{b,c,d\}\)
\(\Rightarrow n[(A\times B)\cap(A\times C)]=?\)
5.
\(A\subset B\)
\(n[(A\times B)\setminus(A\times A)]=7\)
\(\Rightarrow n(B)=?\)
6.
\(A=\{x\mid-1<x<5\quad x\in\mathbb{N}\}\)
\(B=\{y\mid0<y<4\quad y\in\mathbb{N}\}\)
\(\Rightarrow n(A\times B)=?\)