Question

Let A = {□, ◊, ☼, ⌂} and let # be a binary operation from A  A to A presented by the 

following table:

 

# □ ◊ ☼ ⌂

□ □ ◊ ☼ ⌂

◊ ◊ □ ◊ □

☼ ☼ ◊ ☼ ⌂

⌂ ⌂ □ ⌂ ⌂

Answer questions 10 and 11 by referring to the table for #.

Question 10

Which one of the following statements pertaining to the binary operation # is TRUE?

1. ☼ is the identity element for #.

2. # is symmetric (commutative).

3. # is associative.

4. [(⌂ # ◊) # ☼] = [⌂ # (◊ # ☼)]

Question 11

# can be written in list notation. Which one of the following ordered pairs is an element of the list 

notation set representing #?

1. ((□, ◊), ⌂)

2. ((⌂, ☼), ◊)

3. ((☼, ◊), ◊)

4. ((⌂, ◊), ◊)

Answer 1

Question 10

1. Since ☼ # □ = ☼ ≠ □, we conclude that ☼ is not the identity element for #.

Answer: false

2. Taking into account that x # y = y # x for any $x\in A,$ we conclude that the operation # is symmetric (commutative).

Answer: true

3. Since [(⌂ # ◊) # ☼] =□ # ☼ = ☼ but [⌂ # (◊ # ☼)] = ⌂ # ◊ = □ ≠ ☼, the operation # is not associative.

Answer: false

4. Since [(⌂ # ◊) # ☼] =□ # ☼ = ☼ but [⌂ # (◊ # ☼)] = ⌂ # ◊ = □ ≠ ☼, the equality does not hold.

Answer: false

Question 11

1. Since □ # ◊ = ◊ ≠ ⌂, we conclude that ((□, ◊), ⌂) is not an element of the list notation set representing #.

2. Taking into account that ⌂ # ☼ = ⌂ ≠ ◊, we conclude that ((⌂, ☼), ◊) is not an element of the list notation set representing #.

3. Since ☼ # ◊ = ◊, we conclude that ((☼, ◊), ◊) is an element of the list notation set representing #.

4. taking into account that ⌂ # ◊ = □ ≠ ◊, we conclude that ((⌂, ◊), ◊) is not an element of the list notation set representing #.