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A short class with Sarah and her students. The other ways are under More.
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Let's start by defining a symmetric relation. A relation R on a set A is symmetric if, for every a and b in A, if aRb, then bRa. For example, if a and b are people, and R indicates that 'a trusts b', then trust is symmetric.
So, if Alice trusts Bob, it means Bob trusts Alice as well?
Exactly! Now, can anyone think of a relation that is symmetric but not reflexive?
Maybe the relation 'is a sibling of' because not all siblings will consider each other in the same way?
Good example! A symmetric relation can exist without self-relation. Remember: Symmetric means both ways.
Now let's discuss transitive relations. A relation R on a set A is transitive if, whenever aRb and bRc, then aRc as well. Can anyone provide an example?
Like 'is a parent of'? If A is a parent of B, and B is a parent of C, then A must be a grandparent of C?
Exactly right! Remember, transitivity is all about chaining relationships together. It’s not the same as symmetry, where the direction of the relationship matters.
Are there situations when a relation can be both symmetric and transitive?
Yes, a good example is equality. It’s symmetric since if a = b, then b = a, and transitive because if a = b and b = c, then a = c.
Let's prove that symmetric and transitive relations do not necessarily imply reflexivity. Consider the relation R on set X={1, 2} defined as R = {(1, 1), (1, 2), (2, 1)}. Is this relation symmetric?
Yes, it is since it includes (1, 2) and (2, 1).
Good! But is it reflexive?
No, it’s not because (2, 2) isn't included.
Exactly! This shows us that while similarity and transitivity may exist, they do not alone ensure reflexivity.
Now let's look at surjective functions. A function f: A → B is surjective if every element b in B has at least one corresponding a in A. Can we assume surjectivity guarantees bijectivity?
Only if the sets A and B are finite, right?
Correct! A surjective function might not be bijective if B is infinite. Let’s analyze this with a simple example.
An infinite set mapping could lead to non-unique inputs for the same output.
Precisely. This counterexample of surjectivity shows how the relation's nature fundamentally changes with the infinite context.
Symmetry’s a dance we share, trust goes two ways, like a love affair.
Imagine friends on a playground. If Jamie swings with Taylor, then Taylor swings with Jamie — that's symmetry! Meanwhile, if Jamie is on a swing and Taylor is at the slide, and if Taylor's friend Jordan plays with them — that's transitivity in action!
S.T.R.F - Symmetric, Transitive, Reflexive: Such True Relations Fight for fairness.
S is for symmetric, T for transitive, R for reflexive! Remember, not all play nice!
Term
Symmetric Relation
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Term
Transitive Relation
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Term
Surjective Function
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Term
Bijective Function
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A relation R is symmetric if for any a, b in set A, if aRb then bRa.
A relation R is transitive if for any a, b, c in set A, if aRb and bRc, then aRc.
A relation R is reflexive if for every element a in set A, aRa holds true.
A function f: A → B is called surjective if for every b in B, there exists at least one a in A such that f(a) = b.
A function f: A → B is bijective if it is both surjective and injective.
This section discusses symmetric and transitive relations, highlighting their properties and the conditions under which they apply.
The section covers key concepts of symmetric and transitive relations, providing examples and counterexamples, particularly pertaining to reflexivity and the implications of these properties on functions. It emphasizes how certain functions can exhibit these relations under specific conditions.
In this section, we explore the fundamental properties of symmetric and transitive relations, two critical concepts in discrete mathematics. A relation is symmetric if, for any elements a and b, whenever a is related to b, b is also related to a. Transitive relations require that if a is related to b and b is related to c, then a must also be related to c. The importance of these properties becomes clearer through examples, such as showing that a symmetric and transitive relation need not be reflexive. A key point illustrated includes situations where a function can be surjective without being bijective; specifically, it must be finite for bijectivity to hold. We contextualize this with examples of functions mapped within infinite sets and their characteristics, providing a basis for understanding how relations interact in different kinds of sets.