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# Conjunction

In symbolic logic, we can combine two statements to form a compound statement using the word "and." That compound statement is called a conjunction (not to be confused with the part of speech!). Mathematicians prefer to express ideas using numbers and symbols rather than words though, so we don't write "and" to join the statements. Instead, the $\wedge$ symbol is used to denote a conjunction. For example, $p\wedge q$ is a compound statement uniting statements $p$ and $q$ with the word "and."

In this article, we'll explore how to tell if a conjunction is true and look at a few sample statements. Let's begin.

## Working with conjunction

A conjunction is true only if both of the statements in it are true. Let's look at three sample statements to see how this works in practice:

$p$ : A pentagon is a five-sided figure.

$q$ : Red is a color.

$r$ : Wednesday is the last day of the week.

Before we start working with conjunction, it's worth evaluating the truth value of each of these statements individually. Statement $p$ is true because pentagons have five sides. Statement $q$ is true as well since red is a color. However, statement r is false because Wednesday is not regarded as the last day of the week.

Now that we've determined the truth value of each individual statement, we can move on to evaluating the truth value of conjunctions made with them. The conjunction $p\wedge q$ is true because statements $p$ and $q$ are both true independently, so joining them with "and" doesn't change anything. However, the conjunctions $p\wedge r$ and $q\wedge r$ are false because statement $r$ is false. The following truth table may help you visualize when conjunctions are true:

Logic Table

$p$ $q$ $p\wedge q$
T T T
T F F
F T F
F F F

## Conjunction vs. disjunction

A disjunction is similar to a conjunction in that two statements are merged together to form a compound statement. However, the word "or" is used instead of "and," meaning that a disjunction is true if either of its statements is true. In contrast, conjunctions are true only if both statements are true. The word "or" is represented by a $\vee$ symbol in mathematics, so make sure to double-check which symbol you're using in symbolic logic.

## Practice questions on conjunction

a. Using the following statements, evaluate the truth value of $p\wedge q$

p: The formula for the diameter of a circle can be represented as $D=2r$

q: The odds of flipping heads on a fair coin are $\frac{1}{2}$ .

First, we need to check if these statements are true. The diameter of a circle is twice its radius, so statement $p$ is true. The odds of flipping heads on a fair coin are $\frac{1}{2}$ , so statement $q$ is true. Since both statements are true, $p\wedge q$ is also true.

b. Using the following statements, evaluate the truth value of $p\wedge q$ .

p: Jackie Robinson was a professional baseball player.

q: Violet is a primary color.

First, we need to know the truth value of statements $p$ and $q$ . Jackie Robinson is in the Major League Baseball Hall of Fame, so it's true that he was a professional baseball player. However, statement $q$ is false because red, blue, and yellow are the only primary colors.

Next, we have to remember that conjunctions are true only when both statements are true. Since statement $q$ is false, $p\wedge q$ is also false.

c. Consider the following statements:

a: A verb is the action word in a sentence.

b: The Allies won World War I.

c: Mitosis and meiosis are different names for the same thing in biology.

d: ${x}^{2}=2x$ for all real numbers.

Provide an example of both a true conjunction and a false conjunction using the statements above.

The first step here is determining the truth value of the statements we're working with. Statement $a$ is true because verbs are action words. Statement $b$ is also true because the Allies won World War I. However, statement $c$ is false because mitosis and meiosis refer to different processes in biology. Likewise, statement $d$ is false because ${x}^{2}=2x$ is not true for all real numbers.

Next, we have to remember when conjunctions are true. Conjunctions are true if and only if both statements are true. Since we only have two true statements above, the only example of a true conjunction we can write is $a\wedge b$ . Of course, this means that conjunctions are false if at least one of the statements is false. For our false conjunction, we can use $c\wedge d$ , $a\wedge c$ , $a\wedge d$ , $b\wedge c$ , or $b\wedge d$ .

d. Using the following statements, write a conjunction and a disjunction with the same truth value:

$x$ : Rectangles and squares are examples of polygons.

$y$ : 3 is the median of the following set: $\left\{1,2,3,8,28,36\right\}$

$z$ : Right triangles always have an angle measuring 90 degrees.

First, we have to see whether statements $x$ , $y$ , and $z$ are true or false. Statement $x$ is true because rectangles and squares are polygons. Statement $y$ is false because the median would be the mean of 3 and $8=\frac{8+3}{2}=5.5$ . Statement $z$ is true because right triangles always have a right angle measuring 90 degrees.

Next, we have to consider what makes conjunctions and disjunctions true. A conjunction is true only if both statements are true, while disjunctions are true if either statement is true. We want a conjunction and disjunction with the same truth value, which means both have to be true or false. Any conjunction involving statement y will be false, but we cannot write a false disjunction since two of the three statements are true. Therefore, we need a conjunction and disjunction that are both true. One possible combination is $x\wedge z$ and $x\vee z$

## Varsity Tutors can help with conjunction

Conjunctions are simple enough, but many students confuse them with disjunctions or mix up the symbols and end up providing incorrect answers. If you want to help your student avoid these mistakes, a 1-on-1 math tutor can provide additional practice questions until they demonstrate a complete understanding of the distinction between conjunction and disjunction. A private tutor may also provide faster feedback on completed assignments than a classroom teacher with more students. Contact Varsity Tutors today for further information including a personalized quote.

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