Binary operations and their properties Quiz

1. For $a\ast b=a+2b$, find $3\ast4$.

• A. $10$
• B. $11$
• C. $14$
• D. $24$

2. Which statement proves that $a\ast b=a+b+1$ is commutative on the reals?

• A. $1\ast2=4$ and $2\ast1=4$
• B. $a+b+1=b+a+1$ for all $a,b$
• C. $a\ast b$ is a real number
• D. $a\ast0=a+1$

3. For multiplication modulo 7, evaluate $5\otimes4$.

• A. $6$
• B. $20$
• C. $2$
• D. $9$

4. How can you tell from a Cayley table that an operation is commutative?

• A. Every row is different
• B. All entries are in the set
• C. The table is symmetric about the leading diagonal
• D. The leading diagonal is the same

5. For $a\ast b=2a+3b$, which values show that $\ast$ is not associative?

• A. $(1\ast1)\ast1=13$ and $1\ast(1\ast1)=17$
• B. $1\ast2=2\ast1$
• C. $0\ast0=0$
• D. $(0\ast0)\ast0=0\ast(0\ast0)$