What Are the Key Differences in Substitution Notation?

[ver_mathstats]

Homework Statement

We have to do calculations that involve substitutions.

Relevant Equations

(3 · a + 5 · b)[a, b ≔ b, a]
(3 · a + 5 · b)[a ≔ b][b ≔ a]
(3 · a + 5 · b)[b, a ≔ a, b]
(3 · a + 5 · b)[b ≔ a][a ≔ b]

For the first one so far I have
(3 · a + 5 · b)[a, b ≔ b, a]
=⟨ Substitution ⟩
(3 · b + 5 · a)
So far this is right, however I don't really know the difference between the others.

For the second one I did
(3 · a + 5 · b)[a ≔ b][b ≔ a]
=⟨ Substitution ⟩
(3 · b + 5 · b)
For this one I got it wrong, I thought I would have to do two substitutions but it's not working out and I keep getting it wrong. I'm very confused what the difference is between all four.

Any help would be appreciated, thank you.

 

[Mark44]

Homework Statement:: We have to do calculations that involve substitutions.
Relevant Equations:: (3 · a + 5 · b)[a, b ≔ b, a]
(3 · a + 5 · b)[a ≔ b][b ≔ a]
(3 · a + 5 · b)[b, a ≔ a, b]
(3 · a + 5 · b)[b ≔ a][a ≔ b]

What does this notation in these problems mean?

For the first one so far I have
(3 · a + 5 · b)[a, b ≔ b, a]
=⟨ Substitution ⟩
(3 · b + 5 · a)
So far this is right, however I don't really know the difference between the others.

For the second one I did
(3 · a + 5 · b)[a ≔ b][b ≔ a]
=⟨ Substitution ⟩
(3 · b + 5 · b)
For this one I got it wrong, I thought I would have to do two substitutions but it's not working out and I keep getting it wrong. I'm very confused what the difference is between all four.

Any help would be appreciated, thank you.