GATE Overflow Test Series | Discrete Mathematics | Test 1 | Question: 15

Question

Consider the following relations defined on set $I$:

• $R_1:\{(x,y) \mid xy \geq 1 \}$
• $R_2:\{(x,y) \mid x \equiv y(\mod 7) \}$
• $R_3: \{(x,y) \mid x \neq y \}$

where $I$ is set of all non-negative integers. Which of the following statements is TRUE?

• A. $R_1$ reflexive while $R_2$ and $R_3$ are not reflexive.
• B. $R_1$ and $R_3$ are not reflexive while $R_2$ is reflexive.
• C. $R_1,$ $R_2$ and $R_3$ are all reflexive.
• D. None of these relations is reflexive

Comments

for R2 i have a doubt
xRy is reflexive iff xRx && yRy

then if i have 3R10 then to be reflexive 3R3 && 10R10 must be satisfied

here 10R10 isnt satisfied since 10 != 10 mod7

---

I didn’t know that the set of non-negative integers includes 0.

As 0 is neither positive nor negative so I excluded it from the set 🙃

Answer 1

$R_1$ is not reflexive since $(0,0)$ is not included.

$R_2$ relation is equivalent to saying that $(x-y)$ is a multiple of $7$ i.e., $x - y = 7t$ for some integer $t.$ This relation is reflexive, since $x - x = 7 \times 0$ for all $x.$

$R_3$ is not reflexive as none of $(x,x)$ pair is present making it irreflexive.

Comments

x congurent to y mod 5 meaning explained @19:40

https://youtu.be/1nScXLQAQ9A?t=1180

---

Shouldn’t option R1 be correct because here we are updating the domain for the problem and w.r.t that it is Reflexive? 
@Gate Cse @Deepak Poonia @Sachin Mittal 1

---

@Harshit Dubey $R1$ is Not reflexive because $(0,0) \notin R1$.

Answer 2

in R1:   its mentioned that xy>=1   that means we take values of integers whose product gives >=1.  this implies (0,0) cannot be taken initially.   If we take any other pair, they will satisfy the reflexive condition.  eg (-1,3)   then according to reflexive property and given function   -1*-1>=1  and 3*3=9>=1   hence clearly reflexive property is satisfied.

Comments

Then, what's the condition for a relation to be reflexive?