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A short class with Sarah and her students. The other ways are under More.
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Today we'll start with symmetric relations. A relation R on a set S is symmetric if whenever (i,j) is in R, then (j,i) must also be in R. This means the relation is mirrored.
So, if I have (2, 3) in the relation, I need to also have (3, 2)?
Exactly! Think of it as a two-way street; if traffic can go one way, it must be able to go the other. Remember, you can visualize this using pairs.
Are there any specific examples of symmetric relations?
Great question! A common example is 'is a sibling of.' If A is a sibling of B, then B is a sibling of A. Let's remember the acronym SIB for 'Siblings Is a Bi-directional relation.'
Now, moving on to anti-symmetric relations: A relation is anti-symmetric if (a,b) and (b,a) are both in R only if a is equal to b.
So, if I have (1,2) in my relation, I can't have (2,1) unless they are the same?
Correct! That's the essence of anti-symmetry. Let’s use the acronym ANTI: 'A Not TWICE Identical' to remind us that different items can’t be paired both ways.
Does this mean I can have just one of those pairs?
Yes! You can choose either (a,b) or none at all, but not both if a is not equal to b.
Let’s dive into how we count these relations. For symmetric relations over n elements, we can use the formula: 2^{n(n + 1)/2}. Why do you think that is?
Is it because we count half the pairs since they are mirrored?
Exactly! And for anti-symmetric ones, it's a bit different. We have 2^n for the diagonal and 3^{ rac{n(n - 1)}{2}} for other pairs because of three choices—include neither, include one, or both, constrained by equality.
Can you give a quick summary of those formulas?
Sure! Remember the formulas with the acronym C-SAND: Count Symmetric: 2 to the n(n+1)/2, Anti-symmetric: 2^n times 3 to the (n(n-1)/2). It’s a fun way to remember the counting methods!
Let’s look at examples. For symmetric relations, consider an undirected friendship grape where (A,B) implies (B,A). Can anyone think of an anti-symmetric example?
Maybe 'is less than or equal to'? If A is less than B, we can't have B being less than A.
Precisely! That's a perfect example of an anti-symmetric relation. Always remember: less than or equal is directional.
What about equality?
Good point! Equality is both symmetric and anti-symmetric—if a = b, then both (a,b) and (b,a) hold true.
Let's talk about applications. Symmetric relationships are crucial in social networks, while anti-symmetric relations can describe hierarchies. Can someone give an example?
In a company, if two employees are peers, that's a symmetric relationship, but if one is a manager over another, that's anti-symmetric.
Absolutely! Understanding these relationships helps in designing efficient networks and understanding social dynamics.
How about databases?
Great point! Database integrity often uses these properties to maintain relationships between entities, ensuring data consistency.
A relation that holds the property that if (i,j) is present, (j,i) must also be present.
A relation that allows (i,j) and (j,i) Only if i equals j.
Different formulas to determine the number of symmetric and anti-symmetric relations in a set based on its size.
Symmetry’s a mirror, reflecting each side, If (i,j) is present, (j,i) can't hide.
Once in a town, every relationship was strong. Siblings hugged, friends promised to not wrong. But in another part, couples were true, if A and B met, it meant A won’t woo B if he isn’t blue.
Remember SIB for Symmetric Is Both; and ANTI for A Not Twice Identical.
C-SAND for Counting Symmetric and Anti-symmetric: Count Symmetric: 2^(n(n+1)/2), Anti-symmetric: 2^n * 3^(n(n - 1)/2).
Term
Define a symmetric relation.
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Term
Define an anti-symmetric relation.
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A relation where if (i,j) is in R, then (j,i) must also be in R.
A relation where if (a,b) and (b,a) are both in R, then a must be equal to b.
A relation where every element is related to itself.
A relation where no element is related to itself.
The method used to determine the number of possible relations based on the properties defined.
This section explores symmetric and anti-symmetric relations within the context of discrete mathematics, explaining definitions, properties, and counting the number of such relations for given sets.
The section elaborates on the concepts of symmetric and anti-symmetric relations, detailing how symmetric relations require that if (i,j) is in the relation, then (j,i) must also be present, while anti-symmetric relations allow for the presence of (i,j) and (j,i) only if i equals j. The section further outlines methods for counting the number of these relations across a given set of elements.
In discrete mathematics, symmetric and anti-symmetric relations are essential concepts that illustrate how elements within a relation interact with one another.
R over a set S is called symmetric if, for every pair of elements (i, j) in R, its reverse (j, i) also exists in R.(a, b) and (b, a) in R, it must hold that a = b.S of n elements:
2^{n(n+1)/2}.2^n * 3^{(n(n - 1)/2)}.The ability to comprehend and manipulate these relations is pivotal in mathematical logic, data structures, and algorithm design, making this section foundational within discrete mathematics.