📺 Watch Video Explanation:
Given:
\( a * b = a + b + ab, \quad a,b \in \mathbb{R} \setminus \{-1\} \)
1. Closure (Binary Operation Check):
\( a*b = a + b + ab = (1+a)(1+b) – 1 \)
Since \( a \neq -1 \) and \( b \neq -1 \), we have:
\( (1+a)(1+b) \neq 0 \Rightarrow a*b \neq -1 \)
✔ Closed ⇒ Binary operation
2. Commutativity:
\( a*b = a + b + ab = b + a + ba = b*a \)
✔ Commutative
3. Associativity:
\( (a*b)*c = a + b + c + ab + bc + ca + abc \)
\( a*(b*c) = a + b + c + ab + bc + ca + abc \)
✔ Associative
4. Solve \( (2*x)*3 = 7 \):
Step 1:
\( 2*x = 2 + x + 2x = 2 + 3x \)
Step 2:
\( (2*x)*3 = (2+3x) * 3 \)
\( = (2+3x) + 3 + (2+3x)3 \)
\( = 2 + 3x + 3 + 6 + 9x \)
\( = 11 + 12x \)
Step 3:
\( 11 + 12x = 7 \)
\( 12x = -4 \Rightarrow x = -\frac{1}{3} \)
Final Answer:
✔ Binary operation: Yes
✔ Commutative & Associative
✔ \( x = -\frac{1}{3} \)