Question

1. if it is raining, then the concert will be held indoors.

if the concert is held indoors, then the fireworks will be canceled.
∴ if it is raining, then the fireworks will be canceled.
3a let
j: it is raining.
k: the concert will be held indoors.
l: the fireworks will be canceled.
what is the argument in symbolic form?
a
\\(\

$$\begin{array}{c} j \ ightarrow k \\\\ k \ ightarrow l \\\\ \\hline \\therefore \\sim j \ ightarrow \\sim l \\end{array}$$

\\)
b
\\(\

$$\begin{array}{c} j \ ightarrow k \\\\ k \ ightarrow l \\\\ \\hline \\therefore l \ ightarrow j \\end{array}$$

\\)
c
\\(\

$$\begin{array}{c} j \ ightarrow k \\\\ k \\wedge l \\\\ \\hline \\therefore j \\wedge l \\end{array}$$

\\)
d
\\(\

$$\begin{array}{c} j \ ightarrow k \\\\ k \ ightarrow l \\\\ \\hline \\therefore j \ ightarrow l \\end{array}$$

\\)

Explanation:

1. Analyze the given statements:

• The first premise is "If it is raining, then the concert will be held indoors", which is \( j \Rightarrow k \) (since \( j \) is "It is raining" and \( k \) is "The concert will be held indoors").
• The second premise is "If the concert is held indoors, then the fireworks will be canceled", which is \( k \Rightarrow l \) (since \( k \) is "The concert will be held indoors" and \( l \) is "The fireworks will be canceled").
• The conclusion is "If it is raining, then the fireworks will be canceled", which is \( j \Rightarrow l \).

1. Now check each option:

• Option A: The conclusion is \( \sim j\Rightarrow\sim l \), which is the contrapositive of \( j\Rightarrow l \) but not the conclusion we derived, so A is incorrect.
• Option B: The conclusion is \( l\Rightarrow j \), which is the converse of \( j\Rightarrow l \) and not the correct conclusion, so B is incorrect.
• Option C: The second premise is \( k\wedge l \) (which is "The concert is held indoors and the fireworks are canceled") instead of \( k\Rightarrow l \), so C is incorrect.
• Option D: The premises are \( j\Rightarrow k \) and \( k\Rightarrow l \), and the conclusion is \( j\Rightarrow l \), which matches our derivation.

Answer:

D. \(

$$\begin{array}{c}j\Rightarrow k\\k\Rightarrow l\\\hline\therefore j\Rightarrow l\end{array}$$

\)