Question: please respond to as many of the following prompts as...
Please respond to as many of the following prompts as you can by writing readable and valid arguments that exhibit mathematical fluency to the extent that you can do this. please show work clearly and show which answer is for what question. You do not need to answer the second question, I just have it to clear what the third question is about
1. Recall that we discussed number systems by writing an ordered triple (X, Y, Z), where X is a set of things we call `numbers', Y is the notation for an operation we call `addition', and Z is a notation for what we call `multiplication'. We can do something analogous with Logical Systems: we specify the set of statements, the function that determines truth, and the logical operations and operators. For the logical system that we (and essentially everyone) use, let's use the notation (M, Φ, =⇒ , ∧, ∨,¬) to mean that M is the set of statements we work with, Φ is the function that assesses truth, and the others are the logical operations and operator. Please prove or disprove that the our logical system (M, Φ, =⇒ , ∧, ∨,¬) can be replaced with (M, Φ, ∇), where ∇ is defined as follows. For x, y ∈ M, x ∇ y is equivalent to ¬(x ∨ y).please show your work in a truth table format plus a written format.
2. Prove that the following game must have a winner. Place 6 dots equally spaced on a circle on a piece of paper. Player A has a red marker and Player B has a blue marker. Players take turns connecting pairs of dots with line segments using their respective colored markers, only one line segment between any two dots (therefore at most 15 line segments will be drawn). The winner is the first to connect three dots mutually with their respective color, that is, the rest to create a monochromatic \triangle" in their marker's color.
3. Diagram the argument you gave above, using only logical operations and variables such as S1, S2, . . . for the statements made