Overview:
Some of the relationships that are central to algebraic reasoning are also important to logical reasoning. While algebra deals with numbers and variables that are linked together in relationships called equations, logic deals with statements that are linked in relationships called propositions. Just as open sentences can be true or false by their solutions, conditional statements can be true or false.
What Are Conditional Statements?
Conditional statements are also called “if-then” statements. A statement such as “If the sun is shining, then I cast a shadow” has two parts, “the sun is shining” and “I cast a shadow.” Similarly, “If 6 is an even number, then 6 is evenly divisible by 2,” has two parts, “6 is an even number” and “6 is evenly divisible by 2.”
When Is A Conditional Statement True?
A conditional statement is true if both parts of the statement are true. For example, “If the sun is shining” then “I cast a shadow” is true. In addition, a conditional statement is true if both parts of the statement are false. Suppose it is actually dark, so that “the sun is shining” is false. Then, “I cast a shadow” is also false.
When Is a Conditional Statement False?
Suppose the first part of the statement is true, and the sun is shining. However, I didn’t cast a shadow. The statement would be false. Similarly, if x is an even number but x is actually not evenly divisible by 2, the statement is false.
What Is the Converse of a Statement?
The converse of a statement is when the first part of the statement and the second part are reversed. Suppose the statement is “If x is a rose, then x is a flower.” Its converse is “If x is a flower, then x is a rose. Notice that the statement is true because a rose is a flower. However, its converse is not necessarily true. Suppose the flower were a daisy. A daisy is a flower (true), but a daisy is not also a rose.
What Is the Relationship to Equations?
An algebra equation is a statement of conditions in numerical form. In order to find a solution, both parts need to balance. For example, suppose 2x + 6 = 8. In order to make both statements true, 2x = 8 – 6 or 2x = 2, so x = 1. When the equation is checked 2 ∙ 1 + 6 = 8.
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