Use the Law of Detachment and the Law of Syllogism to draw a conclusion from the three given statements - help ASAP please

Question

Use the Law of Detachment and the Law of Syllogism to draw a conclusion from the three given statements.

If it is Friday night, then there is a football game.

If there is a football game, then Josef is wearing his school colors.

It is Friday night.

	If it is Friday night, then Josef is wearing his school colors.
	If it is not Friday night, then Josef is not wearing his school colors.
	Josef is wearing his school colors.
	There is a football game.

Answer 1

(Abbreviations: Law of Detachment = LoD, Law of Syllogism = LoS)

LoD says that "if P then Q" is a true relationship, and P is true, then Q is true. So from

"If it is Friday night, then there is a football game." and "It is Friday night.", LoD gives us "There is a football game.".

LoS says that "if P then Q" is a true relationship, and "if Q then R" is a true relationship, then it must be that "if P then R". So from

"If it is Friday night, then there is a football game." and

"If there is a football game, then Josef is wearing his school colors.", LoS gives us

"If it is Friday night, then Josef is wearing his school colors."

And it's friday, so then LoD gives us "Josef is wearing his school colors."

"If it is not Friday night, then Josef is not wearing his school colors." does not follow from these. Josef can wear his school colors whenever he wants.

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Doing the same thing in a less wordy way: replace written propositions with letters that represent them and can either be true or false. Also write "P=>Q" as the implication proposition "if P then Q is a true relationship".

So I'll define them as follows.

f = It is Friday night.

F = there is a football game.

J = Josef is wearing his school colors.

We are given that

f=>F is true, F=>J is true, and f is true.

LoD says if (P=>Q) is true, and P is true then Q is true.

LoS says if (P=>Q) is true, and (Q=>R), then (P=>R) is true.

So LoD: f=>F and f is true, so F is true (There's a football game).

LoS: f=>F and F=>J so f=>J (Josef wears colors on friday nights).

You can apply LoD to F=>J and F is true, so J is true