Relations

Just basic stuff. Suitable for revision for JEE, IAT, NEST. Question mode: Flashcards only. Answer Mode: Answer with definition. Good luck!

What is the number of elements in the cartesian product of $A$ and $B$ if $A$ has $m$ elements and $B$ has $n$ elements?

$n(A\times B)=m\times n=mn$ elements

What is the number of possible relations for $A\times B$ if $A$ has $m$ elements and $B$ has $n$ elements?

$$2^{mn}$$

In $A\times B$, what is the domain?

$$A$$

In $A\times B$, what is the codomain?

$$B$$

What is a Reflexive Relation?

A relation $R$ on set $A$ is said to be reflexive if every element of A is related to itself.

Thus, $(a,a) \in R$

What is a Symmetric Relation?

A relation $R$ on set $A$ is said to be symmetric if

$$(a,b)\in R \implies (b,a)\in R$$

What is a Transitive Relation?

A relation $R$ on set $A$ is said to be transitive if

$[(a,b)\in R$ and $(b,c)\in R]\implies (a,c)\in R$

What is an Equivalence Relation?

If a relation is reflexive, symmetric, and transitive then it is an equivalence relation

What is an Identity Relation?

A relation is an Identity Relation if

$$I_A=\{(a,a), a\in A\}$$