Question

3. Determine whether the given statements are equivalent.
If a = b, then a•c=b•c.
If a is not equal to b, then a•c is not equal to b•c.​

Answer 1

Step-by-step explanation:

1 Logic

Logical Statements. A logical statement is a mathematical statement that is either

true or false. Here we denote logical statements with capital letters A, B. Logical

statements be combined to form new logical statements as follows:

Name Notation

Conjunction A and B

Disjunction A or B

Negation not A

¬A

Implication A implies B

if A, then B

A ⇒ B

Equivalence A if and only if B

A ⇔ B

Here are some examples of conjunction, disjunction and negation:

x > 1 and x < 3: This is true when x is in the open interval (1, 3).

x > 1 or x < 3: This is true for all real numbers x.

¬(x > 1): This is the same as x ≤ 1.

Here are two logical statements that are true:

x > 4 ⇒ x > 2.

x

2 = 1 ⇔ (x = 1 or x = −1).

Note that “x = 1 or x = −1” is usually written x = ±1.

Converses, Contrapositives, and Tautologies. We begin with converses and

contrapositives:

• The converse of “A implies B” is “B implies A”.

• The contrapositive of “A implies B” is “¬B implies ¬A”

Thus the statement “x > 4 ⇒ x > 2” has:

• Converse: x > 2 ⇒ x > 4.

• Contrapositive: x ≤ 2 ⇒ x ≤ 4.

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