Discussion
ϕ is a function such that 1 ϕ a = 1 and a ϕ b = b ϕ a for all a and b. Which of the following must be true? I. a ϕ 1 = 1 II. (1 ϕ b) ϕ c = 1 ϕ (b ϕ c) III. 
*This question is included in Nova Math - Problem Set D: Defined Functions, question #14
| (A) | I only |
| (B) | ... |
| (C) | ... |
| (D) | ... |
| (E) | ... |
| (F) | ... |
The solution is
Posted: 05/13/2012 19:14
Why does 1(theta) b = 1?
Posted: 05/14/2012 01:55
We are given that
1 ϕ a = 1, or ƒ(1,a) = 1
a ϕ b = b ϕ a, or ƒ(a,b) = ƒ(b,a) which means commutative property
1 ϕ b = ƒ(1,b). By the first definition, it is equal to 1.
1 ϕ a = 1, or ƒ(1,a) = 1
a ϕ b = b ϕ a, or ƒ(a,b) = ƒ(b,a) which means commutative property
1 ϕ b = ƒ(1,b). By the first definition, it is equal to 1.
Posted: 09/10/2012 22:59
I don't understand why II is true....we are not given any info about "c"...?
Posted: 09/08/2013 23:26
Hi Sarah,
In the original solution, we showed that Statement II is true by elimination. Let's also prove it directly:
(1 ϕ b) ϕ c =
1 ϕ c = (We are given that 1 ϕ a = 1, so any letter or number in the position of a will return the value 1. Hence, we replaced 1 ϕ b with 1 in the expression.)
1 (1 ϕ c = 1 for the same reason.)
Now, lets show that the other half of the equation, 1 ϕ (b ϕ c), is also 1. We don't know the values of b and c; but since ϕ is a function, b ϕ c has a value. Let's call that value d. So, b ϕ c = d. Plugging this into the expression 1 ϕ (b ϕ c) yields
1 ϕ (b ϕ c) =
1 ϕ d =
1 (by definition 1 ϕ d = 1 for all values of d.)
Nova Press
In the original solution, we showed that Statement II is true by elimination. Let's also prove it directly:
(1 ϕ b) ϕ c =
1 ϕ c = (We are given that 1 ϕ a = 1, so any letter or number in the position of a will return the value 1. Hence, we replaced 1 ϕ b with 1 in the expression.)
1 (1 ϕ c = 1 for the same reason.)
Now, lets show that the other half of the equation, 1 ϕ (b ϕ c), is also 1. We don't know the values of b and c; but since ϕ is a function, b ϕ c has a value. Let's call that value d. So, b ϕ c = d. Plugging this into the expression 1 ϕ (b ϕ c) yields
1 ϕ (b ϕ c) =
1 ϕ d =
1 (by definition 1 ϕ d = 1 for all values of d.)
Nova Press
Posted: 09/10/2012 23:01
Ohhhh, do we have to assume II is true since both I and III are true?
Posted: 09/03/2013 15:50
The phi symbol seems to represent the cummutative property. (I will use @ to denote Greek Phi.) However, the values of the constants a,b,c are not given and inherently arbitrary. For example, we only know that 1@a = a@1 because a@b=b@a is given. However, we do not know if the symbol represents addition or multiplication because the cummutative property applies to both operations. We should pick one. Lets pick addition for simplicity. If 1@a = 1 then a=0. We still, however, do not know what b is, nor c. It cannot follow that (1@a)/(b@1) certainly equals 1. B could equal five or five million and the @ symbol could equal either multiplication or division. The problem also does not say that a=b which would be necessary for the third definition to follow. The answer to this problem should be I and II only. III is not provable via the information given.
Posted: 09/08/2013 23:54
Hi Nicholas,
We don't know what the function ϕ represents. It could be a mathematical operation other than addition or multiplication, even an operation that we have never seen before.
We do not need a = b for the expression (1ϕa)/(bϕ1) to equal 1. In the proof, we did not assume that a = b. We merely used the definition of ϕ and the Commutative Property.
Nova Press
We don't know what the function ϕ represents. It could be a mathematical operation other than addition or multiplication, even an operation that we have never seen before.
We do not need a = b for the expression (1ϕa)/(bϕ1) to equal 1. In the proof, we did not assume that a = b. We merely used the definition of ϕ and the Commutative Property.
Nova Press
Posted: 09/03/2013 15:51
**Correction: "multiplication or addition."