Answer to Question #89050 in Algebra for RAKESH DEY

Question

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.

Answer 1

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 (a, b) where a ∈ A and b ∈ B. Products can be specified using set-builder notation, e.g.

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

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,

$$\left|A\times B\right|=\left|A\right|\cdot\left|B\right|$$

Similarly,

$$\left|A\times B\times C\right|=\left|A\right|\cdot\left|B\right|\cdot\left|C\right|$$

and so on.

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

In our case,

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

Then,

$$\left|A\times B\times C\right|=\left|A\right|\cdot\left|B\right|\cdot\left|C\right|=1\cdot 2\cdot 2=4$$

Conclusion,

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