Proper Subsets and Relations of Sets

[saaddii]

Q1: Write all proper subsets of S = {1, 2, 3, 4 }.

Q2: Let S = {1,2,5,6 }
Define a relation R on S of at least four order pairs, as (a,b)  R iff a*b is even (i.e. a multiply by b is even)

Q3: Let S = {x, y, z } and R is a relation defined on S such that
R={(y,y),(x,z),(z,x),(x,x),(z,z),(x,y),(y,x)}
Show that R is reflneed proper solution

 

[joypav]

Q1: Write all proper subsets of S = {1, 2, 3, 4 }.

Q2: Let S = {1,2,5,6 }
Define a relation R on S of at least four order pairs, as (a,b)  R iff a*b is even (i.e. a multiply by b is even)

Q3: Let S = {x, y, z } and R is a relation defined on S such that
R={(y,y),(x,z),(z,x),(x,x),(z,z),(x,y),(y,x)}
Show that R is reflneed proper solution

Q1:
A proper subset of $S$ is any subset of $S$ that is not equal to $S$.
So we have...
$\emptyset$
$\left\{1\right\}, \left\{2\right\}, \left\{3\right\}, \left\{4\right\}$
$\left\{1,2\right\}, \left\{1,3\right\}, \left\{1,4\right\}, \left\{2,3\right\}, \left\{2,4\right\}, \left\{3,4\right\}$
$\left\{1,2,3\right\}, \left\{2,3,4\right\}, \left\{1,3,4\right\}, \left\{1,2,4\right\}$

Q3:
R is reflective if for every element $s$ of $S$, $(s,s)$ is in $R$.
So what do you need to check?
For R to be reflexive, it must have elements $(x,x)$, $(y,y)$, and $(z,z)$.