Which of the following relations is not symmetric?

Options

A. `R_(1)` on R defined by `(x,y) in R_(1) iff 1+ xy gt 0` for all `x,y in R`
B. `R_(2)` on `N xx N` defined by `(a,b) R_(2)(c,d)iffa+d=b+c " for all "a, b, c, d in N`
C. `R_(3)` on Z defined by `(a, b) in R_(3) iff b-a` is an even integer
D. `R_(4)` on power set of a set X defined by A `R_(4)` B iff `A sube B`.

Explanation

To determine which of the given relations is not symmetric, we will analyze each relation step by step.

### Step 1: Understand Symmetric Relations
A relation \( R \) on a set is symmetric if for every \( (x, y) \) in \( R \), the pair \( (y, x) \) is also in \( R \).

### Step 2: Analyze Relation \( R_1 \)
The first relation \( R_1 \) is defined as:
\[ (x, y) \in R_1 \iff 1 + xy > 0 \]
To check if this relation is symmetric, assume \( (x, y) \in R_1 \):
\[ 1 + xy > 0 \implies 1 + yx > 0 \quad (\text{since } xy = yx) \]
Thus, \( (y, x) \in R_1 \). Therefore, \( R_1 \) is symmetric.

### Step 3: Analyze Relation \( R_2 \)
The second relation \( R_2 \) is defined as:
\[ (a, b) \in R_2 \iff a + d = b + c \]
This can be rearranged to show:
\[ c + b = d + a \]
If \( (a, b) \in R_2 \), then we can also show that \( (c, d) \in R_2 \). Therefore, \( R_2 \) is symmetric.

### Step 4: Analyze Relation \( R_3 \)
The third relation \( R_3 \) is defined as:
\[ (a, b) \in R_3 \iff b - a \text{ is an even integer} \]
If \( (a, b) \in R_3 \), then \( b - a \) is even. This implies:
\[ a - b \text{ is also even} \]
Thus, \( (b, a) \in R_3 \). Therefore, \( R_3 \) is symmetric.

### Step 5: Analyze Relation \( R_4 \)
The fourth relation \( R_4 \) is defined on the power set \( P(X) \):
\[ (a, b) \in R_4 \iff a \subseteq b \]
If \( (a, b) \in R_4 \), it means \( a \) is a subset of \( b \). However, this does not imply that \( b \) is a subset of \( a \) unless \( a = b \). Therefore, \( R_4 \) is **not symmetric**.

### Conclusion
The relation that is not symmetric is:
\[ \text{Relation } R_4 \]