Here's the answer to your first question:
The first question in the image is asking you to choose the correct symbolic representation of the following argument:
If an equation is of the form y = mx + b, then its graph is a line.
If the graph is not a line, then the equation is not of the form y = mx + b.
To answer this question, you need to know how to use logical symbols and conditional statements. A logical symbol is a notation that represents a logical operation or relation, such as “and”, “or”, “not”, “if…then”, etc. A conditional statement is a statement that has the form “if p, then q”, where p and q are propositions. The proposition p is called the antecedent, and the proposition q is called the consequent.
The argument in the question has two conditional statements:
The first one is “if an equation is of the form y = mx + b, then its graph is a line”. This can be written as p → q, where p is “an equation is of the form y = mx + b” and q is “its graph is a line”.
The second one is “if the graph is not a line, then the equation is not of the form y = mx + b”. This can be written as ¬q → ¬p, where ¬q is “the graph is not a line” and ¬p is “the equation is not of the form y = mx + b”.
The argument can be summarized as:
p → q
¬q → ¬p
This means that p and q are logically equivalent, meaning that they have the same truth value in every possible situation. In other words, if p is true, then q is true, and vice versa; if p is false, then q is false, and vice versa. This logical equivalence can be written as p ↔ q.
Therefore, the correct symbolic representation of the argument is option D: p ↔ q.
Here's the answer to your second question:
To answer this question, you need to know what congruent angles and vertical angles are, and how to use logical symbols to express conditional statements.
Congruent angles are angles that have the same measure. For example, if angle A and angle B are both 45 degrees, then they are congruent.
Vertical angles are angles that are opposite each other when two lines cross. For example, in the image below, angle A and angle C are vertical angles, and angle B and angle D are vertical angles.
!Vertical angles)
A conditional statement is a statement that has the form “if p, then q”, where p and q are propositions. The proposition p is called the antecedent, and the proposition q is called the consequent. A conditional statement can be written using the logical symbol →, which means “implies”. For example, “if it rains, then the grass is wet” can be written as “rains → grass is wet”.
The statement in the question can be written as a conditional statement:
If two angles are congruent, then the angles are vertical angles.
To write this statement using logical symbols, we need to assign a symbol to each proposition. For example, we can use p to represent “two angles are congruent” and q to represent “the angles are vertical angles”. Then, we can write the statement as:
p → q
Therefore, the correct symbolic representation of the statement is option B: p → q.
Here's the answer to your third question:
To answer this question, you need to know how to use logical symbols and modus tollens. A logical symbol is a notation that represents a logical operation or relation, such as “and”, “or”, “not”, “if…then”, etc. Modus tollens is a rule of inference that allows us to draw a valid conclusion from a conditional statement and its negated consequent. The rule can be stated as:
If p, then q.
Not q.
Therefore, not p.
The argument in the question has a conditional statement and its negated consequent:
The conditional statement is “if Brent works this summer, then he will not take a vacation”. This can be written as p → q, where p is “Brent works this summer” and q is “Brent will not take a vacation”.
The negated consequent is “Brent does not work this summer”. This can be written as ¬p, where ¬ means “not”.
The argument can be summarized as:
p → q
¬p
Therefore, ¬q
This means that the argument is an example of modus tollens, and it is valid. To write the argument using logical symbols, we can use the symbol ⊢ to mean “therefore”. For example, “p ⊢ q” means “p therefore q”. Then, we can write the argument as:
p → q, ¬p ⊢ ¬q
Therefore, the correct symbolic representation of the argument is option C: p → q, ¬p ⊢ ¬q.