Question #89050
If a=phi, b={1,2}, c={-1,-2}, then a×b×c has 4 elements.
Is the statement true or false? Justify your answer.
1
Expert's answer
2019-05-14T09:49:01-0400

By the definition,

In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (ab) where a ∈ A and b ∈ B. Products can be specified using set-builder notation, e.g.


A×B={(a,b)aAandbB}A\times B=\left\{(a,b)\left|a\in A\quad and \quad b\in B\right.\right\}

By the definition,

The cardinality of a set is the number of elements of the set. The cardinality of the output set is equal to the product of the cardinalities of all the input sets. That is,


A×B=AB\left|A\times B\right|=\left|A\right|\cdot\left|B\right|

Similarly,


A×B×C=ABC\left|A\times B\times C\right|=\left|A\right|\cdot\left|B\right|\cdot\left|C\right|

and so on.

( More information: https://en.wikipedia.org/wiki/Cartesian_product )

In our case,


{A={ϕ}A=1B={1,2}B=2C={1,2}C=2\left\{\begin{array}{l} A=\left\{\phi\right\}\rightarrow\left|A\right|=1\\ B=\left\{1,2\right\}\rightarrow\left|B\right|=2\\ C=\left\{-1,-2\right\}\rightarrow\left|C\right|=2\\ \end{array}\right.

Then,


A×B×C=ABC=122=4\left|A\times B\times C\right|=\left|A\right|\cdot\left|B\right|\cdot\left|C\right|=1\cdot 2\cdot 2=4

Conclusion,


A×B×Chas4elements.ThestatementisTRUE\boxed{A\times B\times C\,\,\,has\,\,\, 4\,\,\, elements.\,\,\,The\,\,\,statement\,\,\, is\,\,\, TRUE}


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS