The Identity Relation on set X is the set $\lbrace (x, x) | x \in X \rbrace$
Onto vs one to one discrete meaning full#
The Full Relation between sets X and Y is the set $X \times Y$ The Empty Relation between sets X and Y, or on E, is the empty set $\emptyset$ Suppose, there is a relation $R = \lbrace (1, 1), (1,2), (3, 2) \rbrace$ on set $S = \lbrace 1, 2, 3 \rbrace$, it can be represented by the following graph − Types of Relations If there is an ordered pair (x, x), there will be self- loop on vertex ‘x’. For each ordered pair (x, y) in the relation R, there will be a directed edge from the vertex ‘x’ to vertex ‘y’. The number of vertices in the graph is equal to the number of elements in the set from which the relation has been defined. Let, $A = \lbrace 1, 2, 9 \rbrace $ and $ B = \lbrace 1, 3, 7 \rbrace$Ĭase 1 − If relation R is 'equal to' then $R = \lbrace (1, 1), (3, 3) \rbrace$ĭom(R) = $\lbrace 1, 3 \rbrace, Ran(R) = \lbrace 1, 3 \rbrace$Ĭase 2 − If relation R is 'less than' then $R = \lbrace (1, 3), (1, 7), (2, 3), (2, 7) \rbrace$ĭom(R) = $\lbrace 1, 2 \rbrace, Ran(R) = \lbrace 3, 7 \rbrace$Ĭase 3 − If relation R is 'greater than' then $R = \lbrace (2, 1), (9, 1), (9, 3), (9, 7) \rbrace$ĭom(R) = $\lbrace 2, 9 \rbrace, Ran(R) = \lbrace 1, 3, 7 \rbrace$Ī relation can be represented using a directed graph. The range of R, Ran(R), is the set $\lbrace y\: |\: (x, y) \in R \:for\: some\: x\: in\: A\rbrace$
The domain of R, Dom(R), is the set $\lbrace x \:| \: (x, y) \in R \:for\: some\: y\: in\: B \rbrace$ If there are two sets A and B, and relation R have order pair (x, y), then − The minimum cardinality of a relation R is Zero and maximum is $n^2$ in this case.Ī binary relation R on a single set A is a subset of $A \times A$.įor two distinct sets, A and B, having cardinalities m and n respectively, the maximum cardinality of a relation R from A to B is mn. Generally an n-ary relation R between sets $A_1, \dots ,\ and\ A_n$ is a subset of the n-ary product $A_1 \times \dots \times A_n$. If the ordered pair of G is reversed, the relation also changes. Definition and PropertiesĪ binary relation R from set x to y (written as $xRy$ or $R(x,y)$) is a subset of the Cartesian product $x \times y$. Relations may exist between objects of the same set or between objects of two or more sets. (maybe the finite ones are true?)Īny help with these problems would be appreciated.Whenever sets are being discussed, the relationship between the elements of the sets is the next thing that comes up. In part a) I believe that just because a function is one-to-one it does not necessarily have to be onto as can be seen from the diagram.īut then I feel like you can use the same logic to disprove all of them, which I'm not sure it correct. Also that a function is onto if every element in the codomain has a pre-image in the domain (or in other words every element in the codomain must be mapped to an element in the domain).
I know that for a function to be one-to-one there can't be two distinct elements in the domain that map to the same element in the codomain. (d) If $A$ is finite and $f : A \rightarrow A$ is $f$ is onto, then $f$ is one-to-one (c) If $f : A \rightarrow A$ is $f$ is onto, then $f$ is one-to-one.
(b) If $A$ is finite and $f : A \rightarrow A$ is one-to-one, then f is onto. (a) If $f : A \rightarrow A$ is one-to-one, then $f$ is onto. I'm doing some practice problems and am having trouble answering these problems:
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