MHB Operations on Sets: Explained & Examples

  • Thread starter Thread starter bergausstein
  • Start date Start date
  • Tags Tags
    Operations Sets
AI Thread Summary
The discussion clarifies the concept of the Cartesian product of sets, specifically when set A contains a single element. It explains that if A has one element and B has multiple elements, the Cartesian product A × B results in a set of ordered pairs where the first element is from A and the second from B. For example, with A = {a} and B = {b1, b2, b3}, the product A × B yields {(a, b1), (a, b2), (a, b3)}. The thread emphasizes that the total number of pairs is the product of the cardinalities of the sets, confirming that a single element in A leads to as many pairs as there are elements in B. Understanding this operation is essential for grasping more complex mathematical concepts involving sets.
bergausstein
Messages
191
Reaction score
0
please help me understand what my book says:

If set A has only one element a, then $\displaystyle A\,x\,B\,=\, \{\left(a,\, b\right)\,|\,b\,\epsilon\,B\}$, then there is exactly one such element for each element from B.

can you explain what it means and give some examples. thanks! :)
 
Mathematics news on Phys.org
Re: Operations on set

From Wikipedia:

In mathematics, a Cartesian product is a mathematical operation which returns a set (or product set) 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.

For example, if the cardinality of set $A$ is one, where $A=\{a\}$ and we have a set $B$ of cardinality $n$, i.e., $B=\{b_1,b_2,b_3,\cdots,b_n\}$ then:

$$A\,\times\,B=\{(a,b_1),(a,b_2),(a,b_3),\cdots,(a,b_n),\}$$
 
Cartesian products get their name from the prototypical example, the Cartesian plane, which is the Cartesian product of two orthogonal lines.

It's easier to see what is going on if we consider a Cartesian product of two finite sets, say:

A = a bag of red marbles,
B = a bag of green marbles.

Suppose we want "all possible pairs" of marbles, and A has 3 marbles, and B has 4 marbles. We can label these r1,r2,r3 (for the red marbles) and: g1,g2,g3,g4 (for the green marbles). Then the set of all possible pairs looks like this:

(r1,g1) (r1,g2) (r1,g3) (r1,g4)

(r2,g1) (r2,g2) (r2,g3) (r2,g4)

(r3,g1) (r3,g2) (r3,g3) (r3,g4)

Laid out like this, it's clear we have 3*4 = 12 pairs in all. And, in general:

[math]|A \times B| = |A|\cdot|B|[/math]

so, if A and B are sets of 1 element each, their Cartesian product has 1*1 = 1 element (only one possible choice for the "first coordinate", and only one possible choice for the "second coordinate").
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Thread 'Imaginary Pythagoras'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...

Similar threads

Replies
3
Views
2K
Replies
9
Views
2K
Replies
21
Views
6K
Replies
3
Views
2K
Replies
5
Views
2K
Back
Top